Polytope of Type {12,6,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,6,6}*1728c
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 : {12,6,6}*864h
   3-fold quotients : {4,6,6}*576b, {12,6,2}*576a
   4-fold quotients : {6,6,6}*432e
   6-fold quotients : {4,3,6}*288, {4,6,6}*288e, {4,6,6}*288f, {12,6,2}*288d
   9-fold quotients : {4,6,2}*192
   12-fold quotients : {4,3,6}*144, {2,6,6}*144c, {6,6,2}*144a
   18-fold quotients : {4,3,2}*96, {4,6,2}*96b, {4,6,2}*96c
   24-fold quotients : {2,3,6}*72
   36-fold quotients : {4,3,2}*48, {2,6,2}*48, {6,2,2}*48
   72-fold quotients : {2,3,2}*24, {3,2,2}*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)( 13,161)( 14,164)( 15,163)( 16,162)
( 17,157)( 18,160)( 19,159)( 20,158)( 21,165)( 22,168)( 23,167)( 24,166)
( 25,173)( 26,176)( 27,175)( 28,174)( 29,169)( 30,172)( 31,171)( 32,170)
( 33,177)( 34,180)( 35,179)( 36,178)( 37,113)( 38,116)( 39,115)( 40,114)
( 41,109)( 42,112)( 43,111)( 44,110)( 45,117)( 46,120)( 47,119)( 48,118)
( 49,125)( 50,128)( 51,127)( 52,126)( 53,121)( 54,124)( 55,123)( 56,122)
( 57,129)( 58,132)( 59,131)( 60,130)( 61,137)( 62,140)( 63,139)( 64,138)
( 65,133)( 66,136)( 67,135)( 68,134)( 69,141)( 70,144)( 71,143)( 72,142)
( 73,185)( 74,188)( 75,187)( 76,186)( 77,181)( 78,184)( 79,183)( 80,182)
( 81,189)( 82,192)( 83,191)( 84,190)( 85,197)( 86,200)( 87,199)( 88,198)
( 89,193)( 90,196)( 91,195)( 92,194)( 93,201)( 94,204)( 95,203)( 96,202)
( 97,209)( 98,212)( 99,211)(100,210)(101,205)(102,208)(103,207)(104,206)
(105,213)(106,216)(107,215)(108,214);;
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, 
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*s2*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1*s0*s1, 
s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1, 
s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*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)( 13,161)( 14,164)( 15,163)
( 16,162)( 17,157)( 18,160)( 19,159)( 20,158)( 21,165)( 22,168)( 23,167)
( 24,166)( 25,173)( 26,176)( 27,175)( 28,174)( 29,169)( 30,172)( 31,171)
( 32,170)( 33,177)( 34,180)( 35,179)( 36,178)( 37,113)( 38,116)( 39,115)
( 40,114)( 41,109)( 42,112)( 43,111)( 44,110)( 45,117)( 46,120)( 47,119)
( 48,118)( 49,125)( 50,128)( 51,127)( 52,126)( 53,121)( 54,124)( 55,123)
( 56,122)( 57,129)( 58,132)( 59,131)( 60,130)( 61,137)( 62,140)( 63,139)
( 64,138)( 65,133)( 66,136)( 67,135)( 68,134)( 69,141)( 70,144)( 71,143)
( 72,142)( 73,185)( 74,188)( 75,187)( 76,186)( 77,181)( 78,184)( 79,183)
( 80,182)( 81,189)( 82,192)( 83,191)( 84,190)( 85,197)( 86,200)( 87,199)
( 88,198)( 89,193)( 90,196)( 91,195)( 92,194)( 93,201)( 94,204)( 95,203)
( 96,202)( 97,209)( 98,212)( 99,211)(100,210)(101,205)(102,208)(103,207)
(104,206)(105,213)(106,216)(107,215)(108,214);
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, 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*s2*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1*s0*s1, 
s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1, 
s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1 >;

References : None.
to this polytope