Cubical/icosahedral subgroups: lists of elements

Building on the previous three posts, I offer here cubical and icosahedral examples for most of the proper subgroups, specifically for all but the ‘polyaxial’ ones. (Recall here that cubical ‘polyaxial’ subgroups ([3, 3]+, [3, 3], [3+, 4], [4, 3]+) have been presented in the context of 2002-2014 World Cup soccer balls, whereas the only ‘polyaxial’ icosahedral, non-cubical subgroup is the icosahedron’s rotation subgroup, [5, 3]+.) Specifically, I list the isometries in each example — as presented in my Concluding Remarks here — employing the die and soccer ball notations introduced in previous posts, using I for the identity, red color for rotations based on the ‘main axis’ of the ‘monoaxial’ subgroup, green for other (‘free’) rotations, orange for inversion, purple for rotoreflections (by necessity along the ‘main axis’), and blue for reflections. In the spirit of the preceding post, cubical and icosahedral examples for each subgroup type common to the cube and the icosahedron are presented together. There exists precisely one example for each icosahedral subgroup, and up to three algebraically but not geometrically equivalent examples for each cubical subgroup.

[1]+ = {Ι}

[1] = {I, (16)} = {I, (23)(45)} = {Ι, (AG)(DL)(HK)(JI)}

[2+, 2+] = {I, (16)(25)(34)} = {I(AD)(BE)(CF)(GL)(HJ)(IK)}

[2]+ = {I(25)(34)} = {I, (16)(23)(45)} = {I, (AG)(BF)(CE)(DL)(HJ)(IK)}

[2] = {I(25)(34), (25), (34)} = {I, (25)(34)(23)(45), (24)(35)} = {I, (16)(23)(45), (16), (23)(45)} = {I, (AG)(BF)(CE)(DL)(HJ)(IK), (AG)(DL)(HK)(JI), (BF)(CE)(HI)(JK)

[2+, 2] = {I, (25)(34), (16)(25)(34), (16)} = {I, (16)(23)(45), (16)(25)(34), (24)(35)} = {I, (AG)(BF)(CE)(DL)(HJ)(IK), (AD)(BE)(CF)(GL)(HJ)(IK), (AL)(BC)(DG)(EF)}

[2, 2]+ = {I, (25)(34)(16)(23)(45), (16)(24)(35)} = {I, (25)(34)(16)(25), (16)(34)} = {I, (AG)(BF)(CE)(DL)(HJ)(IK), (AD)(BC)(EF)(GL)(HK)(IJ), (AL)(BE)(CF)(DG)(HI)(JK)}

[2, 2] = {I, (25)(34)(16)(23)(45), (16)(24)(35), (16)(25)(34), (16), (24)(35), (23)(45)} = {I, (25)(34)(16)(25), (16)(34), (16)(25)(34), (16), (34), (25)} = {I, (AG)(BF)(CE)(DL)(HJ)(IK), (AD)(BC)(EF)(GL)(HK)(IJ), (AL)(BE)((CF)(DG)(HI)(JK), (AD)(BE)(CF)(GL)(HJ)(IK), (AL)(BC)(DG)(EF), (BF)(CE)(HI)(JK), (AG)(DL)(HK)(IJ)}

[2+, 4+] = {I, (25)(34), (16)(2354), (16)(2453)}

[2+, 4] = {I, (25)(34), (16)(2354), (16)(2453), (16)(23)(45), (16)(24)(35), (25), (34)} = {I, (25)(34), (16)(2354), (16)(2453), (16)(25), (16)(34), (23)(45), (24)(35)}

[3]+ = {I, (123)(465), (132)(456)} = {I(ABG)(CIJ)(DEL)(FKH)(AGB)(CJI)(DLE)(FHK)}

[3] = {I, (123)(465), (132)(456), (12)(56), (13)(46), (23)(45)} = {I, (ABG)(CIJ)(DEL)(FKH), (AGB)(CJI)(DLE)(FHK), (AB)(CJ)(DE)(FH), (AG)(DL)(HK)(IJ), (BG)(CI)(EL)(FK)}

[2, 3]+ = {I(123)(465), (132)(456), (15)(26)(34), (16)(24)(35), (14)(25)(36)} = {I, (ABG)(CIJ)(DEL)(FKH)(AGB)(CJI)(DLE)(FHK), (AD)(BL)(CK)(EG)(FI)(HJ), (AE)(BD)(CH)(FJ)(GL)(IK), (AL)(BE)(CF)(DG)(HI)(JK)}

[2+, 6+] = {I, (123)(465), (132)(456), (142635), (153624),  (16)(25)(34)} = {I(ABG)(CIJ)(DEL)(FKH)(AGB)(CJI)(DLE)(FHK), (ALBDGE)(CHIFJK), (AEGDBL)(CKJFIH)(AD)(BE)(CF)(GL)(HJ)(IK)}

[2+, 6] = {I(123)(465), (132)(456), (142635), (153624)(15)(26)(34), (16)(24)(35), (14)(25)(36), (16)(25)(34), (12)(56), (13)(46), (23)(45)} = {I(ABG)(CIJ)(DEL)(FKH)(AGB)(CJI)(DLE)(FHK), (ALBDGE)(CHIFJK), (AEGDBL)(CKJFIH)(AD)(BL)(CK)(EG)(FI)(HJ), (AE)(BD)(CH)(FJ)(GL)(IK), (AL)(BE)(CF)(DG)(HI)(JK)(AD)(BE)(CF)(GL)(HJ)(IK), (AB)(CJ)(DE)(FH), (AG)(DL)(HK)(IJ), (BG)(CI)(EL)(FK)}

[4]+ = {I, (2354), (2453), (25)(34)}

[4] = {I(2354), (2453), (25)(34), (25), (34), (23)(45), (24)(35)}

[4+, 2] = {I(2354), (2453)(25)(34), (16)(2354), (16)(2453), (16)(25)(34), (16)}

[2, 4]+ = {I(2354), (2453)(25)(34), (16)(25), (16)(34), (16)(23)(45), (16)(24)(35)}

[2, 4] = {I(2354), (2453)(25)(34), (16)(2354), (16)(2453), (16)(25)(34)(16)(25), (16)(34), (16)(23)(45), (16)(24)(35), (16), (25), (34)(23)(45), (24)(35)}

[5]+ = {I, (BGFJK)(CHIEL), (BKJFG)(CLEIH), (BFKGJ)(CILHE), (BJGKF)(CEHLI)}

[5] = {I(BGFJK)(CHIEL), (BKJFG)(CLEIH), (BFKGJ)(CILHE), (BJGKF)(CEHLI), (CH)(GK)(FJ)(IL), (BF)(CE)(HI)(JK), (BK)(EI)(HL)(GJ), (BG)(CI)(EL)(FK). (BJ)(CL)(EH)(FG)}

[2, 5]+ = {I(BGFJK)(CHIEL), (BKJFG)(CLEIH), (BFKGJ)(CILHE), (BJGKF)(CEHLI), (AD)(BC)(EF)(GL)(HK)(IJ), (AD)(BH)(CG)(EJ)(FL)(IK), (AD)(BI)(CF)(EK)(GH)(JL), (AD)(BE)(CJ)(FH)(GI)(KL), (AD)(BL)(CK)(EG)(FI)(HJ)}

[2+, 10+] = {I(BGFJK)(CHIEL), (BKJFG)(CLEIH), (BFKGJ)(CILHE), (BJGKF)(CEHLI), (AD)(BHGIFEJLKC), (AD)(BCKLJEFIGH), (AD)(BLFHKEGCJI), (AD)(BIJCGEKHFL)(AD)(BE)(CF)(GL)(HJ)(IK)}

[2+. 10] = {I(BGFJK)(CHIEL), (BKJFG)(CLEIH), (BFKGJ)(CILHE), (BJGKF)(CEHLI), (AD)(BHGIFEJLKC), (AD)(BCKLJEFIGH), (AD)(BLFHKEGCJI), (AD)(BIJCGEKHFL)(AD)(BC)(EF)(GL)(HK)(IJ), (AD)(BH)(CG)(EJ)(FL)(IK), (AD)(BI)(CF)(EK)(GH)(JL), (AD)(BE)(CJ)(FH)(GI)(KL), (AD)(BL)(CK)(EG)(FI)(HJ)(AD)(BE)(CF)(GL)(HJ)(IK)(CH)(GK)(FJ)(IL), (BF)(CE)(HI)(JK), (BK)(EI)(HL)(GJ), (BG)(CI)(EL)(FK), (BJ)(CL)(EH)(FG)}

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