Polytope of Type {4,24}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,24}*1728c
if this polytope has a name.
Group : SmallGroup(1728,12713)
Rank : 3
Schlafli Type : {4,24}
Number of vertices, edges, etc : 36, 432, 216
Order of s₀s₁s₂ : 24
Order of s₀s₁s₂s₁ : 12
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
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 : {4,12}*864b
   3-fold quotients : {4,24}*576b
   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 : {4,8}*64b
   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)
( 55, 82)( 56, 84)( 57, 83)( 58, 86)( 59, 85)( 60, 87)( 61, 90)( 62, 89)
( 63, 88)( 64,106)( 65,108)( 66,107)( 67,101)( 68,100)( 69,102)( 70,105)
( 71,104)( 72,103)( 73, 95)( 74, 94)( 75, 96)( 76, 99)( 77, 98)( 78, 97)
( 79, 91)( 80, 93)( 81, 92)(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)( 55,175)( 56,176)
( 57,177)( 58,173)( 59,174)( 60,172)( 61,180)( 62,178)( 63,179)( 64,168)
( 65,166)( 66,167)( 67,163)( 68,164)( 69,165)( 70,170)( 71,171)( 72,169)
( 73,185)( 74,186)( 75,184)( 76,183)( 77,181)( 78,182)( 79,187)( 80,188)
( 81,189)( 82,202)( 83,203)( 84,204)( 85,200)( 86,201)( 87,199)( 88,207)
( 89,205)( 90,206)( 91,195)( 92,193)( 93,194)( 94,190)( 95,191)( 96,192)
( 97,197)( 98,198)( 99,196)(100,212)(101,213)(102,211)(103,210)(104,208)
(105,209)(106,214)(107,215)(108,216);;
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*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*s0*s1*s2*s1*s0*s1*s0*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1*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)( 55, 82)( 56, 84)( 57, 83)( 58, 86)( 59, 85)( 60, 87)( 61, 90)
( 62, 89)( 63, 88)( 64,106)( 65,108)( 66,107)( 67,101)( 68,100)( 69,102)
( 70,105)( 71,104)( 72,103)( 73, 95)( 74, 94)( 75, 96)( 76, 99)( 77, 98)
( 78, 97)( 79, 91)( 80, 93)( 81, 92)(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)( 55,175)
( 56,176)( 57,177)( 58,173)( 59,174)( 60,172)( 61,180)( 62,178)( 63,179)
( 64,168)( 65,166)( 66,167)( 67,163)( 68,164)( 69,165)( 70,170)( 71,171)
( 72,169)( 73,185)( 74,186)( 75,184)( 76,183)( 77,181)( 78,182)( 79,187)
( 80,188)( 81,189)( 82,202)( 83,203)( 84,204)( 85,200)( 86,201)( 87,199)
( 88,207)( 89,205)( 90,206)( 91,195)( 92,193)( 93,194)( 94,190)( 95,191)
( 96,192)( 97,197)( 98,198)( 99,196)(100,212)(101,213)(102,211)(103,210)
(104,208)(105,209)(106,214)(107,215)(108,216);
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*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*s0*s1*s2*s1*s0*s1*s0*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1*s0*s1 >;

References : None.
to this polytope