Polytope of Type {12,6,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,6,6}*1728d
if this polytope has a name.
Group : SmallGroup(1728,47874)
Rank : 4
Schlafli Type : {12,6,6}
Number of vertices, edges, etc : 24, 72, 36, 6
Order of s₀s₁s₂s₃ : 6
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {12,3,6}*864
   3-fold quotients : {4,6,6}*576b, {12,6,2}*576b
   4-fold quotients : {6,6,6}*432f
   6-fold quotients : {4,3,6}*288, {4,6,6}*288e, {4,6,6}*288f, {12,3,2}*288
   8-fold quotients : {6,3,6}*216
   9-fold quotients : {4,6,2}*192
   12-fold quotients : {4,3,6}*144, {2,6,6}*144c, {6,6,2}*144b
   18-fold quotients : {4,3,2}*96, {4,6,2}*96b, {4,6,2}*96c
   24-fold quotients : {2,3,6}*72, {6,3,2}*72
   36-fold quotients : {4,3,2}*48, {2,6,2}*48
   72-fold quotients : {2,3,2}*24
   108-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope