Dan Hoey's Taxonomy of Symmetry Groups and Shoenflies Symbols
I have received permission to post Dan Hoey's taxonomy of symmetry groups of Rubik's Cube. Also, I will relate Dan's taxonomy to Shoenflies symbols as implemented in Herbert Kociemba's Cube Explorer. (Go to http://kociemba.org/cube.htm and then click on Symmetric Patterns.) To that end, some preparatory comments are in order.
In order to define any terminology for the symmetry groups of Rubik's Cube, it's necessary first to define some terminology for the symmetries of the cube. To the best of my knowledge, no standard terminology has been adopted by the Rubik's cube community for the symmetries of the cube. The terminology I'm going to use is very similar to some terminology I have seen before, but I can't remember the reference. It may have been Christoph Bandelow's book, Inside Rubik's Cube and Beyond. In any case, if I can find the reference I want to give proper credit.
Cube symmetries are either rotations and reflections. For rotations, the terminology I'm going to use will be based on the axis of rotation for the various symmetries.
A cube contains three kinds of axes of symmetry. It will prove convenient to describe the three kinds of axes in terms of Rubik's Cube faces and cubies.
- Face center to face center axes. There are three such axes: a) center of F face to center of B face, call it the F-B axis, b) center of L face to center of R face, call it the L-R axis, and c) center of U face to center of D face, call it the U-D axis.
- Corner to corner axes. There are four such axes: a) FLU corner to BDR corner, call it the FLU-BRD axis, b) FUR corner to BLD corner, call it the FUR-BLD axis, FRD corner to BUL corner, call it the FRD-BUL axis, and d) FDL corner to BRU corner, call it the FDL-BRU axis.
- Edge to edge axes. There are six such axes: a) FU edge to BD edge, call it the FU-BD axis, b) FR edge to BL edge, call it the FR-BL axis, c) FD edge to BU edge, call it the FD-BU axis, d) FL edge to BR edge, call it the FL-BR axis, e) UL edge to DR edge, call it the UL-DR axis, and f) UR edge to DL edge, call it the UR-DL axis.
We can now define the 24 cube rotations in terms of those three kinds of axes.
Kind of axis | Name of axis | Symmetry name | Conjugacy Class | Description of Symmetry | Symmetry name in Dan Hoey's Taxonomy | Inverse |
---|---|---|---|---|---|---|
none | c[i] | identity | identity | i1 | i1 | |
Kind of axis | Name of axis | Symmetry name | Conjugacy Class | Description of Symmetry | Symmetry name in Dan Hoey's Taxonomy | Inverse |
face center to face center | U-D | c[U'] or c[D] | rotate the entire cube 90° around a face center to face center axis | rotate the entire cube such that the U face is rotated counter-clockwise 90° and the D face is rotated clockwise 90° | q1 | q2 |
face center to face center | U-D | c[U] or c[D'] | rotate the entire cube 90° around a face center to face center axis | rotate the entire cube such that the U face is rotated clockwise 90° and the D face is rotated counter-clockwise 90° | q2 | q1 |
face center to face center | F-B | c[F'] or c[B] | rotate the entire cube 90° around a face center to face center axis | rotate the entire cube such that the F face is rotated counter-clockwise 90° and the B face is rotated clockwise 90° | q3 | q4 |
face center to face center | F-B | c[F] or c[B'] | rotate the entire cube 90° around a face center to face center axis | rotate the entire cube such that the F face is rotated clockwise 90° and the B face is rotated counter-clockwise 90° | q4 | q3 |
face center to face center | L-R | c[L] or c[R'] | rotate the entire cube 90° around a face center to face center axis | rotate the entire cube such that the L face is rotated clockwise 90° and the R face is rotated counter-clockwise 90° | q5 | q6 |
face center to face center | L-R | c[L'] or c[R] | rotate the entire cube 90° around a face center to face center axis | rotate the entire cube such that the L face is rotated counter-clockwise 90° and the R face is rotated clockwise 90° | q6 | q5 |
Kind of axis | Name of axis | Symmetry name | Conjugacy Class | Description of Symmetry | Symmetry name in Dan Hoey's Taxonomy | Inverse |
face center to face center | U-D | c[U2] or c[D2] | rotate the entire cube 180° around a face center to face center axis | rotate the entire cube such that the U face is rotated 180° and the D face is rotated 180° | k1 | k1 |
face center to face center | F-B | c[F2] or c[B2] | rotate the entire cube 180° around a face center to face center axis | rotate the entire cube such that the F face is rotated 180° and the B face is rotated 180° | k2 | k2 |
face center to face center | L-R | c[L2] or c[R2] | rotate the entire cube 180° around a face center to face center axis | rotate the entire cube such that the L face is rotated 180° and the R face is rotated 180° | k3 | k3 |
Kind of axis | Name of axis | Symmetry name | Conjugacy Class | Description of Symmetry | Symmetry name in Dan Hoey's Taxonomy | Inverse |
corner to corner | ULB-DFR | c[ULB] or c[DRF] | rotate the entire cube 120° around a corner to corner axis | rotate the entire cube such that the ULB cubie is rotated clockwise 120° and the DFR cubie is rotated counterclockwise 120° | g1 | g2 |
corner to corner | ULB-DFR | c[UBL] or c[DFR] | rotate the entire cube 120° around a corner to corner axis | rotate the entire cube such that the ULB cubie is rotated counterclockwise 120° and the DFR cubie is rotated clockwise 120° | g2 | g1 |
corner to corner | DBL-URF | c[DBL] or c[UFR] | rotate the entire cube 120° around a corner to corner axis | rotate the entire cube such that the DBL cubie is rotated clockwise 120° and the URF cubie is rotated counterclockwise 120° | g3 | g4 |
corner to corner | DBL-URF | c[DLB] or c[URF] | rotate the entire cube 120° around a corner to corner axis | rotate the entire cube such that the DBL cubie is rotated counterclockwise 120° and the URF cubie is rotated clockwise 120° | g4 | g3 |
corner to corner | DLF-UBR | c[DLF] or c[URB] | rotate the entire cube 120° around a corner to corner axis | rotate the entire cube such that the DLF cubie is rotated clockwise 120° and the URB cubie is rotated counterclockwise 120° | g5 | g6 |
corner to corner | DLF-UBR | c[DFL] or c[UBR] | rotate the entire cube 120° around a corner to corner axis | rotate the entire cube such that the DLF cubie is rotated counterclockwise 120° and the UBR cubie is rotated clockwise 120° | g6 | g5 |
corner to corner | UFL-DRB | c[ULF] or c[DRB] | rotate the entire cube 120° around a corner to corner axis | rotate the entire cube such that the UFL cubie is rotated counterclockwise 120° and the DRB cubie is rotated clockwise 120° | g7 | g8 |
corner to corner | UFL-DRB | c[UFL] or c[DBR] | rotate the entire cube 120° around a corner to corner axis | rotate the entire cube such that the UFL cubie is rotated clockwise 120° and the DRB cubie is rotated counterclockwise 120° | g8 | g7 |
Kind of axis | Name of axis | Symmetry name | Conjugacy Class | Description of Symmetry | Symmetry name in Dan Hoey's Taxonomy | Inverse |
edge to edge | FL-BR | c[FL] or c[BR] | rotate the entire cube 180° around an edge to edge axis | rotate the entire cube such that the FL cubie is rotated 180° and the BR cubie is rotated 180° | d1 | d1 |
edge to edge | FR-BL | c[FR] or c[BL] | rotate the entire cube 180° around an edge to edge axis | rotate the entire cube such that the FR cubie is rotated 180° and the BL cubie is rotated 180° | d2 | d2 |
edge to edge | UL-DR | c[UL] or c[DR] | rotate the entire cube 180° around an edge to edge axis | rotate the entire cube such that the UL cubie is rotated 180° and the DR cubie is rotated 180° | d3 | d3 |
edge to edge | UR-DL | c[UR] or c[DL] | rotate the entire cube 180° around an edge to edge axis | rotate the entire cube such that the UR cubie is rotated 180° and the DL cubie is rotated 180° | d4 | d4 |
edge to edge | FU-BD | c[FU] or c[BD] | rotate the entire cube 180° around an edge to edge axis | rotate the entire cube such that the FU cubie is rotated 180° and the BD cubie is rotated 180° | d5 | d5 |
edge to edge | FD-BU | c[FD] or c[BU] | rotate the entire cube 180° around an edge to edge axis | rotate the entire cube such that the FD cubie is rotated 180° and the BU cubie is rotated 180° | d6 | d6 |
For the reflections, I'm going to arrive at Dan Hoey's taxonomy simply by multiplying all the rotations by a reflection. If C is the set of 24 rotations and if f is any reflection, then it is the case that fC=Cf. The choice of f is arbitrary. However, it is in general not the case that fc=cf for every element c of C. It is not necessary for fc=cf for all c in C in order that fC=Cf. However, it is convenient to choose f such that fc=cf for all c in C. There is only one such reflection that commutes with all the rotations, namely the central inversion that we will denote as v. Hence, we have vC=Cv, and we also have cv=vc for all c in C. In Dan Hoey's taxonomy, the convention of generating the reflections as vC or Cv yields the following.
l1=v(i1) r1=v(q1) j1=v(k1) h1=v(g1) e1=v(d1) r2=v(q2) j2=v(k2) h2=v(g2) e2=v(d2) r3=v(q3) j3=v(k3) h3=v(g3) e3=v(d3) r4=v(q4) h4=v(g4) e4=v(d4) r5=v(q5) h5=v(g5) e5=v(d5) r6=v(q6) h6=v(g6) e6=v(d6) h7=v(g7) h8=v(g8)
The following table lists the 98 subgroups of M, and also the corresponding 33 symmetry classes. The subgroups are arranged by symmetry class, and the symmetry class name is given both in terms of Dan Hoey's taxonomy, and in terms of Shoenflies symbols. The subgroups within any particular symmetry class are respectively conjugate subgroups of each other.
I1 = {i1} I = {I1} Shoenflies symbol = C1 HV1 = {i1,l1} HV = {HV1} Shoenflies symbol = Ci AV1 = {i1,e1} AV2 = {i1,e2} AV3 = {i1,e3} AV4 = {i1,e4} AV5 = {i1,e5} AV6 = {i1,e6} AV = {AV1,AV2,AV3,AV4,AV5,AV6} Shoenflies symbol = Cs(b) CV1 = {i1,d1} CV2 = {i1,d2} CV3 = {i1,d3} CV4 = {i1,d4} CV5 = {i1,d5} CV6 = {i1,d6} CV = {CV1,CV2,CV3,CV4,CV5,CV6} Shoenflies symbol = C2(b) HW1 = {i1,j1} HW2 = {i1,j2} HW3 = {i1,j3} HW = {HW1,HW2,HW3} Shoenflies symbol = Cs(a) ES1 = {i1,k1} ES2 = {i1,k2} ES3 = {i1,k3} ES={ES1,ES2,ES3} Shoenflies symbol = C2(a) ET1 = {i1,g1,g2} ET2 = {i1,g3,g4} ET3 = {i1,g5,g6} ET4 = {i1,g7,g8} ET= = {ET1,ET2,ET3,ET4} Shoenflies symbol = C3 V1 = {i1,l1,e1,d1} V2 = {i1,l1,e2,d2} V3 = {i1,l1,e3,d3} V4 = {i1,l1,e4,d4} V5 = {i1,l1,e5,d5} V6 = {i1,l1,e6,d6} V = {V1,V2,V3,V4,V5,V6} Shoenflies symbol = C2h(b) W1 = {i1,j1,e1,d2} W2 = {i1,j1,e2,d1} W3 = {i1,j2,e3,d4} W4 = {i1,j2,e4,d3} W5 = {i1,j3,e5,d6} W6 = {i1,j3,e6,d5} W = {W1,W2,W3,W4,W5,W6} Shoenflies symbol = C2v(b) HS1 = {i1,l1,k1,j1} HS2 = {i1,l1,k2,j2} HS3 = {i1,l1,k3,j3} HS = {HS1,HS2,HS3} Shoenflies symbol = C2h(a) AS1 = {i1,k1,e1,e2} AS2 = {i1,k2,e3,e4} AS3 = {i1,k3,e5,e6} AS = {AS1,AS2,AS3} Shoenflies symbol = C2h(a) CR1 = {i1,k1,q1,q2} CR2 = {i1,k2,q3,q4} CR3 = {i1,k3,q5,q6} CR = {CR1,CR2,CR3} Shoenflies symbol = C4 HQ1 = {i1,k1,j2,j3} HQ2 = {i1,k2,j1,j3} HQ3 = {i1,k3,j1,j2} HQ = {HQ1,HQ2,HQ3} Shoenflies symbol = C2v(a2) AP1 = {i1,k1,r1,r2} AP2 = {i1,k2,r3,r4} AP3 = {i1,k3,r5,r6} AP = {AP1,AP2,AP3} Shoenflies symbol = S4 CS1 = {i1,k1,d1,d2} CS2 = {i1,k2,d3,d4} CS3 = {i1,k3,d5,d6} CS = {CS1,CS2,CS3} Shoenflies symbol = D2 (edge) EX1 = {i1,k1,k2,k3} EX = {EX1} Shoenflies symbol = D2 (face) HT1 = {i1,l1,g1,g2,h1,h2} HT2 = {i1,l1,g3,g4,h3,h4} HT3 = {i1,l1,g5,g6,h5,h6} HT4 = {i1,l1,g7,g8,h7,h8} HT = {HT1,HT2,HT3,HT4} Shoenflies symbol = S6 AT1 = {i1,e1,e4,e5,g1,g2} AT2 = {i1,e2,e3,e5,g3,g4} AT3 = {i1,e1,e3,e6,g5,g6} AT4 = {i1,e2,e4,e6,g7,g8} AT = {AT1,AT2,AT3,AT4} Shoenflies symbol = C3v CT1 = {i1,d1,d4,d5,g1,g2} CT2 = {i1,d2,d3,d5,g3,g4} CT3 = {i1,d1,d3,d6,g5,g6} CT4 = {i1,d2,d4,d6,g7,g8} CT = {CT1,CT2,CT3,CT4} Shoenflies symbol = D3 S1 = {i1,l1,k1,j1,e1,e2,d1,d2} S2 = {i1,l1,k2,j2,e3,e4,d3,d4} S3 = {i1,l1,k3,j3,e5,e6,d5,d6} S = {S1,S2,S3} Shoenflies symbol = D2h (edge) Q1 = {i1,k1,j2,j3,e1,e2,q1,q2} Q2 = {i1,k2,j1,j3,e3,e4,q3,q4} Q3 = {i1,k3,j1,j2,e5,e6,q5,q6} Q = {Q1,Q2,Q3} Shoenflies symbol = C4v R1 = {i1,l1,k1,j1,q1,q2,r1,r2} R2 = {i1,l1,k2,j2,q3,q4,r3,r4} R3 = {i1,l1,k3,j3,q5,q6,r5,r6} R = {R1,R2,R3} Shoenflies Symbol = C4h P1 = {i1,k1,j2,j3,d1,d2,r1,r2} P2 = {i1,k2,j1,j3,d3,d4,r3,r4} P3 = {i1,k3,j1,j2,d5,d6,r5,r6} P = {P1,P2,P3} Shoenflies Symbol = D2d (edge) HX1 = {i1,l1,k1,k2,k3,j1,j2,j3} HX = {HX1} Shoenflies Symbol = D2h (face) AX1 = {i1,k1,k2,k3,e1,e2,r1,r2} AX2 = {i1,k1,k2,k3,e3,e4,r3,r4} AX3 = {i1,k1,k2,k3,e5,e6,r5,r6} AX = {AX1,AX2,AX3} Shoenflies Symbol = D2d (face) CX1 = {i1,k1,k2,k3,d1,d2,q1,q2} CX2 = {i1,k1,k2,k3,d3,d4,q3,q4} CX3 = {i1,k1,k2,k3,d5,d6,q5,q6} CX = {CX1,CX2,CX3} Shoenflies Symbol = D4 T1 = {i1,l1,e1,e4,e5,d1,d4,d5,g1,g2,h1,h2} T2 = {i1,l1,e2,e3,e5,d2,d3,d5,g3,g4,h3,h4} T3 = {i1,l1,e1,e3,e6,d1,d3,d6,g5,g6,h5,h6} T4 = {i1,l1,e2,e4,e6,d2,d4,d6,g7,g8,h7,h8} T = {T1,T2,T3,T4} Shoenflies Symbol = D3d E1 = {i1,k1,k2,k3,g1,g2,g3,g4,g5,g6,g7,g8} E = {E1} Shoenflies Symbol = T X1 = {i1,l1,k1,k2,k3,j1,j2,j3,e1,e2,d1,d2,q1,q2,r1,r2} X2 = {i1,l1,k1,k2,k3,j1,j2,j3,e3,e4,d3,d4,q3,q4,r3,r4} X3 = {i1,l1,k1,k2,k3,j1,j2,j3,e5,e6,d5,d6,q5,q6,r5,r6} X = {X1,X2,X3} Shoenflies Symbol = D4h H1 = {i1,l1,k1,k2,k3,j1,j2,j3,g1,g2,g3,g4,g5,g6,g7,g8, h1,h2,h3,h4,h5,h6,h7,h8} H = {H1} Shoenflies Symbol = Th A1 = {i1,k1,k2,k3,e1,e2,e3,e4,e5,e6,r1,r2,r3,r4,r5,r6, g1,g2,g3,g4,g5,g6,g7,g8} A = {A1} Shoenflies Symbol = Td C1 = {i1,k1,k2,k3,d1,d2,d3,d4,d5,d6,q1,q2,q3,q4,q5,q6, g1,g2,g3,g4,g5,g6,g7,g8} C = {C1} Shoenflies Symbol = O M1 = {i1,l1,k1,k2,k3,j1,j2,j3,e1,e2,e3,e4,e5,e6, d1,d2,d3,d4,d5,d6,q1,q2,q3,q4,q5,q6, r1,r2,r3,r4,r5,r6,g1,g2,g3,g4,g5,g6,g7,g8, h1,h2,h3,h4,h5,h6,h7,h8} M = {M1} Shoenflies Symbol = Oh