Polytope of Type {24,12}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {24,12}*1728i
if this polytope has a name.
Group : SmallGroup(1728,12703)
Rank : 3
Schlafli Type : {24,12}
Number of vertices, edges, etc : 72, 432, 36
Order of s₀s₁s₂ : 8
Order of s₀s₁s₂s₁ : 6
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 : {12,12}*864f
   3-fold quotients : {24,4}*576a
   4-fold quotients : {6,12}*432f
   6-fold quotients : {12,4}*288
   8-fold quotients : {6,12}*216b
   12-fold quotients : {6,4}*144
   24-fold quotients : {6,4}*72
   27-fold quotients : {8,4}*64a
   54-fold quotients : {4,4}*32, {8,2}*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 := (  1,109)(  2,110)(  3,111)(  4,117)(  5,115)(  6,116)(  7,113)(  8,114)
(  9,112)( 10,129)( 11,127)( 12,128)( 13,134)( 14,135)( 15,133)( 16,130)
( 17,131)( 18,132)( 19,119)( 20,120)( 21,118)( 22,124)( 23,125)( 24,126)
( 25,123)( 26,121)( 27,122)( 28,136)( 29,137)( 30,138)( 31,144)( 32,142)
( 33,143)( 34,140)( 35,141)( 36,139)( 37,156)( 38,154)( 39,155)( 40,161)
( 41,162)( 42,160)( 43,157)( 44,158)( 45,159)( 46,146)( 47,147)( 48,145)
( 49,151)( 50,152)( 51,153)( 52,150)( 53,148)( 54,149)( 55,190)( 56,191)
( 57,192)( 58,198)( 59,196)( 60,197)( 61,194)( 62,195)( 63,193)( 64,210)
( 65,208)( 66,209)( 67,215)( 68,216)( 69,214)( 70,211)( 71,212)( 72,213)
( 73,200)( 74,201)( 75,199)( 76,205)( 77,206)( 78,207)( 79,204)( 80,202)
( 81,203)( 82,163)( 83,164)( 84,165)( 85,171)( 86,169)( 87,170)( 88,167)
( 89,168)( 90,166)( 91,183)( 92,181)( 93,182)( 94,188)( 95,189)( 96,187)
( 97,184)( 98,185)( 99,186)(100,173)(101,174)(102,172)(103,178)(104,179)
(105,180)(106,177)(107,175)(108,176);;
s1 := (  1,  4)(  2,  6)(  3,  5)(  8,  9)( 10, 18)( 11, 17)( 12, 16)( 13, 15)
( 19, 21)( 22, 27)( 23, 26)( 24, 25)( 28, 31)( 29, 33)( 30, 32)( 35, 36)
( 37, 45)( 38, 44)( 39, 43)( 40, 42)( 46, 48)( 49, 54)( 50, 53)( 51, 52)
( 55, 85)( 56, 87)( 57, 86)( 58, 82)( 59, 84)( 60, 83)( 61, 88)( 62, 90)
( 63, 89)( 64, 99)( 65, 98)( 66, 97)( 67, 96)( 68, 95)( 69, 94)( 70, 93)
( 71, 92)( 72, 91)( 73,102)( 74,101)( 75,100)( 76,108)( 77,107)( 78,106)
( 79,105)( 80,104)( 81,103)(109,166)(110,168)(111,167)(112,163)(113,165)
(114,164)(115,169)(116,171)(117,170)(118,180)(119,179)(120,178)(121,177)
(122,176)(123,175)(124,174)(125,173)(126,172)(127,183)(128,182)(129,181)
(130,189)(131,188)(132,187)(133,186)(134,185)(135,184)(136,193)(137,195)
(138,194)(139,190)(140,192)(141,191)(142,196)(143,198)(144,197)(145,207)
(146,206)(147,205)(148,204)(149,203)(150,202)(151,201)(152,200)(153,199)
(154,210)(155,209)(156,208)(157,216)(158,215)(159,214)(160,213)(161,212)
(162,211);;
s2 := (  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)
( 56, 57)( 58, 76)( 59, 78)( 60, 77)( 61, 72)( 62, 71)( 63, 70)( 64, 73)
( 65, 75)( 66, 74)( 68, 69)( 79, 81)( 83, 84)( 85,103)( 86,105)( 87,104)
( 88, 99)( 89, 98)( 90, 97)( 91,100)( 92,102)( 93,101)( 95, 96)(106,108)
(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)
(164,165)(166,184)(167,186)(168,185)(169,180)(170,179)(171,178)(172,181)
(173,183)(174,182)(176,177)(187,189)(191,192)(193,211)(194,213)(195,212)
(196,207)(197,206)(198,205)(199,208)(200,210)(201,209)(203,204)(214,216);;
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*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2*s1, 
s2*s0*s1*s2*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s0*s1, 
s2*s0*s1*s0*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1*s0*s1*s0*s1*s2*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, 
s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;

Permutation Representation (Magma) :

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

References : None.
to this polytope