Polytope of Type {4,12}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,12}*1728b
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
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {4,12}*864a
   3-fold quotients : {4,4}*576
   4-fold quotients : {4,12}*432a
   6-fold quotients : {4,4}*288
   8-fold quotients : {4,12}*216
   12-fold quotients : {4,4}*144
   24-fold quotients : {4,4}*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)(  2,150)(  3,149)(  4,156)(  5,155)(  6,154)(  7,142)(  8,144)
(  9,143)( 10,158)( 11,157)( 12,159)( 13,136)( 14,138)( 15,137)( 16,152)
( 17,151)( 18,153)( 19,141)( 20,140)( 21,139)( 22,146)( 23,145)( 24,147)
( 25,162)( 26,161)( 27,160)( 28,121)( 29,123)( 30,122)( 31,129)( 32,128)
( 33,127)( 34,115)( 35,117)( 36,116)( 37,131)( 38,130)( 39,132)( 40,109)
( 41,111)( 42,110)( 43,125)( 44,124)( 45,126)( 46,114)( 47,113)( 48,112)
( 49,119)( 50,118)( 51,120)( 52,135)( 53,134)( 54,133)( 55,175)( 56,177)
( 57,176)( 58,183)( 59,182)( 60,181)( 61,169)( 62,171)( 63,170)( 64,185)
( 65,184)( 66,186)( 67,163)( 68,165)( 69,164)( 70,179)( 71,178)( 72,180)
( 73,168)( 74,167)( 75,166)( 76,173)( 77,172)( 78,174)( 79,189)( 80,188)
( 81,187)( 82,202)( 83,204)( 84,203)( 85,210)( 86,209)( 87,208)( 88,196)
( 89,198)( 90,197)( 91,212)( 92,211)( 93,213)( 94,190)( 95,192)( 96,191)
( 97,206)( 98,205)( 99,207)(100,195)(101,194)(102,193)(103,200)(104,199)
(105,201)(106,216)(107,215)(108,214);;
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*s1*s2*s1*s2*s0*s1*s2*s1*s2*s1*s2*s1, 
s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s2*s1*s0*s2*s1*s2*s1*s2*s1*s0, 
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1 ];;
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)(  2,150)(  3,149)(  4,156)(  5,155)(  6,154)(  7,142)
(  8,144)(  9,143)( 10,158)( 11,157)( 12,159)( 13,136)( 14,138)( 15,137)
( 16,152)( 17,151)( 18,153)( 19,141)( 20,140)( 21,139)( 22,146)( 23,145)
( 24,147)( 25,162)( 26,161)( 27,160)( 28,121)( 29,123)( 30,122)( 31,129)
( 32,128)( 33,127)( 34,115)( 35,117)( 36,116)( 37,131)( 38,130)( 39,132)
( 40,109)( 41,111)( 42,110)( 43,125)( 44,124)( 45,126)( 46,114)( 47,113)
( 48,112)( 49,119)( 50,118)( 51,120)( 52,135)( 53,134)( 54,133)( 55,175)
( 56,177)( 57,176)( 58,183)( 59,182)( 60,181)( 61,169)( 62,171)( 63,170)
( 64,185)( 65,184)( 66,186)( 67,163)( 68,165)( 69,164)( 70,179)( 71,178)
( 72,180)( 73,168)( 74,167)( 75,166)( 76,173)( 77,172)( 78,174)( 79,189)
( 80,188)( 81,187)( 82,202)( 83,204)( 84,203)( 85,210)( 86,209)( 87,208)
( 88,196)( 89,198)( 90,197)( 91,212)( 92,211)( 93,213)( 94,190)( 95,192)
( 96,191)( 97,206)( 98,205)( 99,207)(100,195)(101,194)(102,193)(103,200)
(104,199)(105,201)(106,216)(107,215)(108,214);
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*s1*s2*s1*s2*s0*s1*s2*s1*s2*s1*s2*s1, 
s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s2*s1*s0*s2*s1*s2*s1*s2*s1*s0, 
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1 >;

References : None.
to this polytope