Polytope of Type {4,12}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,12}*1728a
if this polytope has a name.
Group : SmallGroup(1728,12630)
Rank : 3
Schlafli Type : {4,12}
Number of vertices, edges, etc : 72, 432, 216
Order of s₀s₁s₂ : 12
Order of s₀s₁s₂s₁ : 12
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
   Halving Operation
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {4,12}*864b
   3-fold quotients : {4,12}*576
   4-fold quotients : {4,6}*432a
   6-fold quotients : {4,12}*288
   8-fold quotients : {4,6}*216
   12-fold quotients : {4,6}*144
   24-fold quotients : {4,6}*72
   27-fold quotients : {4,4}*64
   54-fold quotients : {4,4}*32
   108-fold quotients : {2,4}*16, {4,2}*16
   216-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Finitely Presented Group Representation (GAP) :

F := FreeGroup("s0","s1","s2");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  
rels := [ s0*s0, s1*s1, s2*s2, s0*s2*s0*s2, s0*s1*s0*s1*s0*s1*s0*s1, 
s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1, 
s2*s0*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;

Permutation Representation (Magma) :

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

Finitely Presented Group Representation (Magma) :

poly<s0,s1,s2> := Group< s0,s1,s2 | s0*s0, s1*s1, s2*s2, 
s0*s2*s0*s2, s0*s1*s0*s1*s0*s1*s0*s1, 
s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1, 
s2*s0*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;

References : None.
to this polytope