# Polytope of Type {6,12,3}

Atlas Canonical Name : {6,12,3}*864a
if this polytope has a name.
Group : SmallGroup(864,4000)
Rank : 4
Schlafli Type : {6,12,3}
Number of vertices, edges, etc : 6, 72, 36, 6
Order of s0s1s2s3 : 6
Order of s0s1s2s3s2s1 : 6
Special Properties :
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{6,12,3,2} of size 1728
Vertex Figure Of :
{2,6,12,3} of size 1728
Quotients (Maximal Quotients in Boldface) :
3-fold quotients : {6,4,3}*288
4-fold quotients : {6,6,3}*216a
8-fold quotients : {3,6,3}*108
9-fold quotients : {2,4,3}*96
12-fold quotients : {6,2,3}*72
18-fold quotients : {2,4,3}*48
24-fold quotients : {3,2,3}*36
36-fold quotients : {2,2,3}*24
Covers (Minimal Covers in Boldface) :
2-fold covers : {12,12,3}*1728a, {6,24,3}*1728a, {6,12,6}*1728a
Permutation Representation (GAP) :
s0 := (  5,  9)(  6, 10)(  7, 11)(  8, 12)( 13, 25)( 14, 26)( 15, 27)( 16, 28)
( 17, 33)( 18, 34)( 19, 35)( 20, 36)( 21, 29)( 22, 30)( 23, 31)( 24, 32)
( 41, 45)( 42, 46)( 43, 47)( 44, 48)( 49, 61)( 50, 62)( 51, 63)( 52, 64)
( 53, 69)( 54, 70)( 55, 71)( 56, 72)( 57, 65)( 58, 66)( 59, 67)( 60, 68)
( 77, 81)( 78, 82)( 79, 83)( 80, 84)( 85, 97)( 86, 98)( 87, 99)( 88,100)
( 89,105)( 90,106)( 91,107)( 92,108)( 93,101)( 94,102)( 95,103)( 96,104)
(113,117)(114,118)(115,119)(116,120)(121,133)(122,134)(123,135)(124,136)
(125,141)(126,142)(127,143)(128,144)(129,137)(130,138)(131,139)(132,140)
(149,153)(150,154)(151,155)(152,156)(157,169)(158,170)(159,171)(160,172)
(161,177)(162,178)(163,179)(164,180)(165,173)(166,174)(167,175)(168,176)
(185,189)(186,190)(187,191)(188,192)(193,205)(194,206)(195,207)(196,208)
(197,213)(198,214)(199,215)(200,216)(201,209)(202,210)(203,211)(204,212);;
s1 := (  1,127)(  2,128)(  3,125)(  4,126)(  5,123)(  6,124)(  7,121)(  8,122)
(  9,131)( 10,132)( 11,129)( 12,130)( 13,115)( 14,116)( 15,113)( 16,114)
( 17,111)( 18,112)( 19,109)( 20,110)( 21,119)( 22,120)( 23,117)( 24,118)
( 25,139)( 26,140)( 27,137)( 28,138)( 29,135)( 30,136)( 31,133)( 32,134)
( 33,143)( 34,144)( 35,141)( 36,142)( 37,163)( 38,164)( 39,161)( 40,162)
( 41,159)( 42,160)( 43,157)( 44,158)( 45,167)( 46,168)( 47,165)( 48,166)
( 49,151)( 50,152)( 51,149)( 52,150)( 53,147)( 54,148)( 55,145)( 56,146)
( 57,155)( 58,156)( 59,153)( 60,154)( 61,175)( 62,176)( 63,173)( 64,174)
( 65,171)( 66,172)( 67,169)( 68,170)( 69,179)( 70,180)( 71,177)( 72,178)
( 73,199)( 74,200)( 75,197)( 76,198)( 77,195)( 78,196)( 79,193)( 80,194)
( 81,203)( 82,204)( 83,201)( 84,202)( 85,187)( 86,188)( 87,185)( 88,186)
( 89,183)( 90,184)( 91,181)( 92,182)( 93,191)( 94,192)( 95,189)( 96,190)
( 97,211)( 98,212)( 99,209)(100,210)(101,207)(102,208)(103,205)(104,206)
(105,215)(106,216)(107,213)(108,214);;
s2 := (  2,  3)(  5,  9)(  6, 11)(  7, 10)(  8, 12)( 14, 15)( 17, 21)( 18, 23)
( 19, 22)( 20, 24)( 26, 27)( 29, 33)( 30, 35)( 31, 34)( 32, 36)( 37, 73)
( 38, 75)( 39, 74)( 40, 76)( 41, 81)( 42, 83)( 43, 82)( 44, 84)( 45, 77)
( 46, 79)( 47, 78)( 48, 80)( 49, 85)( 50, 87)( 51, 86)( 52, 88)( 53, 93)
( 54, 95)( 55, 94)( 56, 96)( 57, 89)( 58, 91)( 59, 90)( 60, 92)( 61, 97)
( 62, 99)( 63, 98)( 64,100)( 65,105)( 66,107)( 67,106)( 68,108)( 69,101)
( 70,103)( 71,102)( 72,104)(110,111)(113,117)(114,119)(115,118)(116,120)
(122,123)(125,129)(126,131)(127,130)(128,132)(134,135)(137,141)(138,143)
(139,142)(140,144)(145,181)(146,183)(147,182)(148,184)(149,189)(150,191)
(151,190)(152,192)(153,185)(154,187)(155,186)(156,188)(157,193)(158,195)
(159,194)(160,196)(161,201)(162,203)(163,202)(164,204)(165,197)(166,199)
(167,198)(168,200)(169,205)(170,207)(171,206)(172,208)(173,213)(174,215)
(175,214)(176,216)(177,209)(178,211)(179,210)(180,212);;
s3 := (  1, 73)(  2, 76)(  3, 75)(  4, 74)(  5, 81)(  6, 84)(  7, 83)(  8, 82)
(  9, 77)( 10, 80)( 11, 79)( 12, 78)( 13, 93)( 14, 96)( 15, 95)( 16, 94)
( 17, 89)( 18, 92)( 19, 91)( 20, 90)( 21, 85)( 22, 88)( 23, 87)( 24, 86)
( 25,101)( 26,104)( 27,103)( 28,102)( 29, 97)( 30,100)( 31, 99)( 32, 98)
( 33,105)( 34,108)( 35,107)( 36,106)( 38, 40)( 41, 45)( 42, 48)( 43, 47)
( 44, 46)( 49, 57)( 50, 60)( 51, 59)( 52, 58)( 54, 56)( 61, 65)( 62, 68)
( 63, 67)( 64, 66)( 70, 72)(109,181)(110,184)(111,183)(112,182)(113,189)
(114,192)(115,191)(116,190)(117,185)(118,188)(119,187)(120,186)(121,201)
(122,204)(123,203)(124,202)(125,197)(126,200)(127,199)(128,198)(129,193)
(130,196)(131,195)(132,194)(133,209)(134,212)(135,211)(136,210)(137,205)
(138,208)(139,207)(140,206)(141,213)(142,216)(143,215)(144,214)(146,148)
(149,153)(150,156)(151,155)(152,154)(157,165)(158,168)(159,167)(160,166)
(162,164)(169,173)(170,176)(171,175)(172,174)(178,180);;
poly := Group([s0,s1,s2,s3]);;

Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s2*s0*s2,
s0*s3*s0*s3, s1*s3*s1*s3, s2*s3*s2*s3*s2*s3,
s2*s0*s1*s2*s1*s2*s0*s1*s2*s1, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s2*s0*s3*s2*s1*s0*s1*s2*s0*s3*s1*s0*s1*s2*s0*s1*s0*s1 ];;
poly := F / rels;;

Permutation Representation (Magma) :
s0 := Sym(216)!(  5,  9)(  6, 10)(  7, 11)(  8, 12)( 13, 25)( 14, 26)( 15, 27)
( 16, 28)( 17, 33)( 18, 34)( 19, 35)( 20, 36)( 21, 29)( 22, 30)( 23, 31)
( 24, 32)( 41, 45)( 42, 46)( 43, 47)( 44, 48)( 49, 61)( 50, 62)( 51, 63)
( 52, 64)( 53, 69)( 54, 70)( 55, 71)( 56, 72)( 57, 65)( 58, 66)( 59, 67)
( 60, 68)( 77, 81)( 78, 82)( 79, 83)( 80, 84)( 85, 97)( 86, 98)( 87, 99)
( 88,100)( 89,105)( 90,106)( 91,107)( 92,108)( 93,101)( 94,102)( 95,103)
( 96,104)(113,117)(114,118)(115,119)(116,120)(121,133)(122,134)(123,135)
(124,136)(125,141)(126,142)(127,143)(128,144)(129,137)(130,138)(131,139)
(132,140)(149,153)(150,154)(151,155)(152,156)(157,169)(158,170)(159,171)
(160,172)(161,177)(162,178)(163,179)(164,180)(165,173)(166,174)(167,175)
(168,176)(185,189)(186,190)(187,191)(188,192)(193,205)(194,206)(195,207)
(196,208)(197,213)(198,214)(199,215)(200,216)(201,209)(202,210)(203,211)
(204,212);
s1 := Sym(216)!(  1,127)(  2,128)(  3,125)(  4,126)(  5,123)(  6,124)(  7,121)
(  8,122)(  9,131)( 10,132)( 11,129)( 12,130)( 13,115)( 14,116)( 15,113)
( 16,114)( 17,111)( 18,112)( 19,109)( 20,110)( 21,119)( 22,120)( 23,117)
( 24,118)( 25,139)( 26,140)( 27,137)( 28,138)( 29,135)( 30,136)( 31,133)
( 32,134)( 33,143)( 34,144)( 35,141)( 36,142)( 37,163)( 38,164)( 39,161)
( 40,162)( 41,159)( 42,160)( 43,157)( 44,158)( 45,167)( 46,168)( 47,165)
( 48,166)( 49,151)( 50,152)( 51,149)( 52,150)( 53,147)( 54,148)( 55,145)
( 56,146)( 57,155)( 58,156)( 59,153)( 60,154)( 61,175)( 62,176)( 63,173)
( 64,174)( 65,171)( 66,172)( 67,169)( 68,170)( 69,179)( 70,180)( 71,177)
( 72,178)( 73,199)( 74,200)( 75,197)( 76,198)( 77,195)( 78,196)( 79,193)
( 80,194)( 81,203)( 82,204)( 83,201)( 84,202)( 85,187)( 86,188)( 87,185)
( 88,186)( 89,183)( 90,184)( 91,181)( 92,182)( 93,191)( 94,192)( 95,189)
( 96,190)( 97,211)( 98,212)( 99,209)(100,210)(101,207)(102,208)(103,205)
(104,206)(105,215)(106,216)(107,213)(108,214);
s2 := Sym(216)!(  2,  3)(  5,  9)(  6, 11)(  7, 10)(  8, 12)( 14, 15)( 17, 21)
( 18, 23)( 19, 22)( 20, 24)( 26, 27)( 29, 33)( 30, 35)( 31, 34)( 32, 36)
( 37, 73)( 38, 75)( 39, 74)( 40, 76)( 41, 81)( 42, 83)( 43, 82)( 44, 84)
( 45, 77)( 46, 79)( 47, 78)( 48, 80)( 49, 85)( 50, 87)( 51, 86)( 52, 88)
( 53, 93)( 54, 95)( 55, 94)( 56, 96)( 57, 89)( 58, 91)( 59, 90)( 60, 92)
( 61, 97)( 62, 99)( 63, 98)( 64,100)( 65,105)( 66,107)( 67,106)( 68,108)
( 69,101)( 70,103)( 71,102)( 72,104)(110,111)(113,117)(114,119)(115,118)
(116,120)(122,123)(125,129)(126,131)(127,130)(128,132)(134,135)(137,141)
(138,143)(139,142)(140,144)(145,181)(146,183)(147,182)(148,184)(149,189)
(150,191)(151,190)(152,192)(153,185)(154,187)(155,186)(156,188)(157,193)
(158,195)(159,194)(160,196)(161,201)(162,203)(163,202)(164,204)(165,197)
(166,199)(167,198)(168,200)(169,205)(170,207)(171,206)(172,208)(173,213)
(174,215)(175,214)(176,216)(177,209)(178,211)(179,210)(180,212);
s3 := Sym(216)!(  1, 73)(  2, 76)(  3, 75)(  4, 74)(  5, 81)(  6, 84)(  7, 83)
(  8, 82)(  9, 77)( 10, 80)( 11, 79)( 12, 78)( 13, 93)( 14, 96)( 15, 95)
( 16, 94)( 17, 89)( 18, 92)( 19, 91)( 20, 90)( 21, 85)( 22, 88)( 23, 87)
( 24, 86)( 25,101)( 26,104)( 27,103)( 28,102)( 29, 97)( 30,100)( 31, 99)
( 32, 98)( 33,105)( 34,108)( 35,107)( 36,106)( 38, 40)( 41, 45)( 42, 48)
( 43, 47)( 44, 46)( 49, 57)( 50, 60)( 51, 59)( 52, 58)( 54, 56)( 61, 65)
( 62, 68)( 63, 67)( 64, 66)( 70, 72)(109,181)(110,184)(111,183)(112,182)
(113,189)(114,192)(115,191)(116,190)(117,185)(118,188)(119,187)(120,186)
(121,201)(122,204)(123,203)(124,202)(125,197)(126,200)(127,199)(128,198)
(129,193)(130,196)(131,195)(132,194)(133,209)(134,212)(135,211)(136,210)
(137,205)(138,208)(139,207)(140,206)(141,213)(142,216)(143,215)(144,214)
(146,148)(149,153)(150,156)(151,155)(152,154)(157,165)(158,168)(159,167)
(160,166)(162,164)(169,173)(170,176)(171,175)(172,174)(178,180);
poly := sub<Sym(216)|s0,s1,s2,s3>;

Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2,
s3*s3, s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3,
s2*s3*s2*s3*s2*s3, s2*s0*s1*s2*s1*s2*s0*s1*s2*s1,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s2*s0*s3*s2*s1*s0*s1*s2*s0*s3*s1*s0*s1*s2*s0*s1*s0*s1 >;

References : None.
to this polytope