Polytope of Type {24,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {24,4}*1728d
if this polytope has a name.
Group : SmallGroup(1728,12713)
Rank : 3
Schlafli Type : {24,4}
Number of vertices, edges, etc : 216, 432, 36
Order of s₀s₁s₂ : 24
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 : {12,4}*864a
   3-fold quotients : {8,4}*576b
   4-fold quotients : {12,4}*432a
   6-fold quotients : {4,4}*288
   8-fold quotients : {12,4}*216
   12-fold quotients : {4,4}*144
   24-fold quotients : {4,4}*72
   27-fold quotients : {8,4}*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 := (  1,136)(  2,138)(  3,137)(  4,151)(  5,153)(  6,152)(  7,159)(  8,158)
(  9,157)( 10,146)( 11,145)( 12,147)( 13,161)( 14,160)( 15,162)( 16,139)
( 17,141)( 18,140)( 19,156)( 20,155)( 21,154)( 22,144)( 23,143)( 24,142)
( 25,149)( 26,148)( 27,150)( 28,109)( 29,111)( 30,110)( 31,124)( 32,126)
( 33,125)( 34,132)( 35,131)( 36,130)( 37,119)( 38,118)( 39,120)( 40,134)
( 41,133)( 42,135)( 43,112)( 44,114)( 45,113)( 46,129)( 47,128)( 48,127)
( 49,117)( 50,116)( 51,115)( 52,122)( 53,121)( 54,123)( 55,190)( 56,192)
( 57,191)( 58,205)( 59,207)( 60,206)( 61,213)( 62,212)( 63,211)( 64,200)
( 65,199)( 66,201)( 67,215)( 68,214)( 69,216)( 70,193)( 71,195)( 72,194)
( 73,210)( 74,209)( 75,208)( 76,198)( 77,197)( 78,196)( 79,203)( 80,202)
( 81,204)( 82,163)( 83,165)( 84,164)( 85,178)( 86,180)( 87,179)( 88,186)
( 89,185)( 90,184)( 91,173)( 92,172)( 93,174)( 94,188)( 95,187)( 96,189)
( 97,166)( 98,168)( 99,167)(100,183)(101,182)(102,181)(103,171)(104,170)
(105,169)(106,176)(107,175)(108,177);;
s1 := (  1,  2)(  4,  8)(  5,  7)(  6,  9)( 10, 13)( 11, 15)( 12, 14)( 17, 18)
( 19, 25)( 20, 27)( 21, 26)( 23, 24)( 28, 29)( 31, 35)( 32, 34)( 33, 36)
( 37, 40)( 38, 42)( 39, 41)( 44, 45)( 46, 52)( 47, 54)( 48, 53)( 50, 51)
( 55, 83)( 56, 82)( 57, 84)( 58, 89)( 59, 88)( 60, 90)( 61, 86)( 62, 85)
( 63, 87)( 64, 94)( 65, 96)( 66, 95)( 67, 91)( 68, 93)( 69, 92)( 70, 97)
( 71, 99)( 72, 98)( 73,106)( 74,108)( 75,107)( 76,103)( 77,105)( 78,104)
( 79,100)( 80,102)( 81,101)(109,191)(110,190)(111,192)(112,197)(113,196)
(114,198)(115,194)(116,193)(117,195)(118,202)(119,204)(120,203)(121,199)
(122,201)(123,200)(124,205)(125,207)(126,206)(127,214)(128,216)(129,215)
(130,211)(131,213)(132,212)(133,208)(134,210)(135,209)(136,164)(137,163)
(138,165)(139,170)(140,169)(141,171)(142,167)(143,166)(144,168)(145,175)
(146,177)(147,176)(148,172)(149,174)(150,173)(151,178)(152,180)(153,179)
(154,187)(155,189)(156,188)(157,184)(158,186)(159,185)(160,181)(161,183)
(162,182);;
s2 := (  1, 20)(  2, 19)(  3, 21)(  4, 13)(  5, 15)(  6, 14)(  7,  8)( 10, 11)
( 16, 26)( 17, 25)( 18, 27)( 23, 24)( 28, 47)( 29, 46)( 30, 48)( 31, 40)
( 32, 42)( 33, 41)( 34, 35)( 37, 38)( 43, 53)( 44, 52)( 45, 54)( 50, 51)
( 55,101)( 56,100)( 57,102)( 58, 94)( 59, 96)( 60, 95)( 61, 89)( 62, 88)
( 63, 90)( 64, 92)( 65, 91)( 66, 93)( 67, 85)( 68, 87)( 69, 86)( 70,107)
( 71,106)( 72,108)( 73, 83)( 74, 82)( 75, 84)( 76,103)( 77,105)( 78,104)
( 79, 98)( 80, 97)( 81, 99)(109,128)(110,127)(111,129)(112,121)(113,123)
(114,122)(115,116)(118,119)(124,134)(125,133)(126,135)(131,132)(136,155)
(137,154)(138,156)(139,148)(140,150)(141,149)(142,143)(145,146)(151,161)
(152,160)(153,162)(158,159)(163,209)(164,208)(165,210)(166,202)(167,204)
(168,203)(169,197)(170,196)(171,198)(172,200)(173,199)(174,201)(175,193)
(176,195)(177,194)(178,215)(179,214)(180,216)(181,191)(182,190)(183,192)
(184,211)(185,213)(186,212)(187,206)(188,205)(189,207);;
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, s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s0*s1*s0*s1*s0*s1*s0*s1*s2*s0*s1*s0*s1*s0*s1*s0*s1, 
s2*s0*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s0*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1 ];;
poly := F / rels;;

Permutation Representation (Magma) :

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

References : None.
to this polytope