Polytope of Type {24,6,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {24,6,6}*1728d
if this polytope has a name.
Group : SmallGroup(1728,33799)
Rank : 4
Schlafli Type : {24,6,6}
Number of vertices, edges, etc : 24, 72, 18, 6
Order of s₀s₁s₂s₃ : 24
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Universal
   Orientable
   Flat
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 : {12,6,6}*864d
   3-fold quotients : {24,2,6}*576, {24,6,2}*576b
   4-fold quotients : {6,6,6}*432g
   6-fold quotients : {24,2,3}*288, {12,2,6}*288, {12,6,2}*288b
   8-fold quotients : {3,6,6}*216b
   9-fold quotients : {24,2,2}*192, {8,2,6}*192
   12-fold quotients : {12,2,3}*144, {6,2,6}*144, {6,6,2}*144c
   18-fold quotients : {8,2,3}*96, {12,2,2}*96, {4,2,6}*96
   24-fold quotients : {3,2,6}*72, {3,6,2}*72, {6,2,3}*72
   27-fold quotients : {8,2,2}*64
   36-fold quotients : {4,2,3}*48, {2,2,6}*48, {6,2,2}*48
   48-fold quotients : {3,2,3}*36
   54-fold quotients : {4,2,2}*32
   72-fold quotients : {2,2,3}*24, {3,2,2}*24
   108-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := (  2,  3)(  4,  7)(  5,  9)(  6,  8)( 11, 12)( 13, 16)( 14, 18)( 15, 17)
( 20, 21)( 22, 25)( 23, 27)( 24, 26)( 29, 30)( 31, 34)( 32, 36)( 33, 35)
( 38, 39)( 40, 43)( 41, 45)( 42, 44)( 47, 48)( 49, 52)( 50, 54)( 51, 53)
( 55, 82)( 56, 84)( 57, 83)( 58, 88)( 59, 90)( 60, 89)( 61, 85)( 62, 87)
( 63, 86)( 64, 91)( 65, 93)( 66, 92)( 67, 97)( 68, 99)( 69, 98)( 70, 94)
( 71, 96)( 72, 95)( 73,100)( 74,102)( 75,101)( 76,106)( 77,108)( 78,107)
( 79,103)( 80,105)( 81,104)(109,163)(110,165)(111,164)(112,169)(113,171)
(114,170)(115,166)(116,168)(117,167)(118,172)(119,174)(120,173)(121,178)
(122,180)(123,179)(124,175)(125,177)(126,176)(127,181)(128,183)(129,182)
(130,187)(131,189)(132,188)(133,184)(134,186)(135,185)(136,190)(137,192)
(138,191)(139,196)(140,198)(141,197)(142,193)(143,195)(144,194)(145,199)
(146,201)(147,200)(148,205)(149,207)(150,206)(151,202)(152,204)(153,203)
(154,208)(155,210)(156,209)(157,214)(158,216)(159,215)(160,211)(161,213)
(162,212);;
s1 := (  1,113)(  2,112)(  3,114)(  4,110)(  5,109)(  6,111)(  7,116)(  8,115)
(  9,117)( 10,122)( 11,121)( 12,123)( 13,119)( 14,118)( 15,120)( 16,125)
( 17,124)( 18,126)( 19,131)( 20,130)( 21,132)( 22,128)( 23,127)( 24,129)
( 25,134)( 26,133)( 27,135)( 28,140)( 29,139)( 30,141)( 31,137)( 32,136)
( 33,138)( 34,143)( 35,142)( 36,144)( 37,149)( 38,148)( 39,150)( 40,146)
( 41,145)( 42,147)( 43,152)( 44,151)( 45,153)( 46,158)( 47,157)( 48,159)
( 49,155)( 50,154)( 51,156)( 52,161)( 53,160)( 54,162)( 55,194)( 56,193)
( 57,195)( 58,191)( 59,190)( 60,192)( 61,197)( 62,196)( 63,198)( 64,203)
( 65,202)( 66,204)( 67,200)( 68,199)( 69,201)( 70,206)( 71,205)( 72,207)
( 73,212)( 74,211)( 75,213)( 76,209)( 77,208)( 78,210)( 79,215)( 80,214)
( 81,216)( 82,167)( 83,166)( 84,168)( 85,164)( 86,163)( 87,165)( 88,170)
( 89,169)( 90,171)( 91,176)( 92,175)( 93,177)( 94,173)( 95,172)( 96,174)
( 97,179)( 98,178)( 99,180)(100,185)(101,184)(102,186)(103,182)(104,181)
(105,183)(106,188)(107,187)(108,189);;
s2 := (  4,  7)(  5,  8)(  6,  9)( 10, 19)( 11, 20)( 12, 21)( 13, 25)( 14, 26)
( 15, 27)( 16, 22)( 17, 23)( 18, 24)( 31, 34)( 32, 35)( 33, 36)( 37, 46)
( 38, 47)( 39, 48)( 40, 52)( 41, 53)( 42, 54)( 43, 49)( 44, 50)( 45, 51)
( 58, 61)( 59, 62)( 60, 63)( 64, 73)( 65, 74)( 66, 75)( 67, 79)( 68, 80)
( 69, 81)( 70, 76)( 71, 77)( 72, 78)( 85, 88)( 86, 89)( 87, 90)( 91,100)
( 92,101)( 93,102)( 94,106)( 95,107)( 96,108)( 97,103)( 98,104)( 99,105)
(112,115)(113,116)(114,117)(118,127)(119,128)(120,129)(121,133)(122,134)
(123,135)(124,130)(125,131)(126,132)(139,142)(140,143)(141,144)(145,154)
(146,155)(147,156)(148,160)(149,161)(150,162)(151,157)(152,158)(153,159)
(166,169)(167,170)(168,171)(172,181)(173,182)(174,183)(175,187)(176,188)
(177,189)(178,184)(179,185)(180,186)(193,196)(194,197)(195,198)(199,208)
(200,209)(201,210)(202,214)(203,215)(204,216)(205,211)(206,212)(207,213);;
s3 := (  1, 10)(  2, 11)(  3, 12)(  4, 13)(  5, 14)(  6, 15)(  7, 16)(  8, 17)
(  9, 18)( 28, 37)( 29, 38)( 30, 39)( 31, 40)( 32, 41)( 33, 42)( 34, 43)
( 35, 44)( 36, 45)( 55, 64)( 56, 65)( 57, 66)( 58, 67)( 59, 68)( 60, 69)
( 61, 70)( 62, 71)( 63, 72)( 82, 91)( 83, 92)( 84, 93)( 85, 94)( 86, 95)
( 87, 96)( 88, 97)( 89, 98)( 90, 99)(109,118)(110,119)(111,120)(112,121)
(113,122)(114,123)(115,124)(116,125)(117,126)(136,145)(137,146)(138,147)
(139,148)(140,149)(141,150)(142,151)(143,152)(144,153)(163,172)(164,173)
(165,174)(166,175)(167,176)(168,177)(169,178)(170,179)(171,180)(190,199)
(191,200)(192,201)(193,202)(194,203)(195,204)(196,205)(197,206)(198,207);;
poly := Group([s0,s1,s2,s3]);;

Finitely Presented Group Representation (GAP) :

F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s2*s0*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s1*s2*s3*s2*s1*s2*s3*s2, 
s2*s0*s1*s2*s1*s2*s0*s1*s2*s1, s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(216)!(  2,  3)(  4,  7)(  5,  9)(  6,  8)( 11, 12)( 13, 16)( 14, 18)
( 15, 17)( 20, 21)( 22, 25)( 23, 27)( 24, 26)( 29, 30)( 31, 34)( 32, 36)
( 33, 35)( 38, 39)( 40, 43)( 41, 45)( 42, 44)( 47, 48)( 49, 52)( 50, 54)
( 51, 53)( 55, 82)( 56, 84)( 57, 83)( 58, 88)( 59, 90)( 60, 89)( 61, 85)
( 62, 87)( 63, 86)( 64, 91)( 65, 93)( 66, 92)( 67, 97)( 68, 99)( 69, 98)
( 70, 94)( 71, 96)( 72, 95)( 73,100)( 74,102)( 75,101)( 76,106)( 77,108)
( 78,107)( 79,103)( 80,105)( 81,104)(109,163)(110,165)(111,164)(112,169)
(113,171)(114,170)(115,166)(116,168)(117,167)(118,172)(119,174)(120,173)
(121,178)(122,180)(123,179)(124,175)(125,177)(126,176)(127,181)(128,183)
(129,182)(130,187)(131,189)(132,188)(133,184)(134,186)(135,185)(136,190)
(137,192)(138,191)(139,196)(140,198)(141,197)(142,193)(143,195)(144,194)
(145,199)(146,201)(147,200)(148,205)(149,207)(150,206)(151,202)(152,204)
(153,203)(154,208)(155,210)(156,209)(157,214)(158,216)(159,215)(160,211)
(161,213)(162,212);
s1 := Sym(216)!(  1,113)(  2,112)(  3,114)(  4,110)(  5,109)(  6,111)(  7,116)
(  8,115)(  9,117)( 10,122)( 11,121)( 12,123)( 13,119)( 14,118)( 15,120)
( 16,125)( 17,124)( 18,126)( 19,131)( 20,130)( 21,132)( 22,128)( 23,127)
( 24,129)( 25,134)( 26,133)( 27,135)( 28,140)( 29,139)( 30,141)( 31,137)
( 32,136)( 33,138)( 34,143)( 35,142)( 36,144)( 37,149)( 38,148)( 39,150)
( 40,146)( 41,145)( 42,147)( 43,152)( 44,151)( 45,153)( 46,158)( 47,157)
( 48,159)( 49,155)( 50,154)( 51,156)( 52,161)( 53,160)( 54,162)( 55,194)
( 56,193)( 57,195)( 58,191)( 59,190)( 60,192)( 61,197)( 62,196)( 63,198)
( 64,203)( 65,202)( 66,204)( 67,200)( 68,199)( 69,201)( 70,206)( 71,205)
( 72,207)( 73,212)( 74,211)( 75,213)( 76,209)( 77,208)( 78,210)( 79,215)
( 80,214)( 81,216)( 82,167)( 83,166)( 84,168)( 85,164)( 86,163)( 87,165)
( 88,170)( 89,169)( 90,171)( 91,176)( 92,175)( 93,177)( 94,173)( 95,172)
( 96,174)( 97,179)( 98,178)( 99,180)(100,185)(101,184)(102,186)(103,182)
(104,181)(105,183)(106,188)(107,187)(108,189);
s2 := Sym(216)!(  4,  7)(  5,  8)(  6,  9)( 10, 19)( 11, 20)( 12, 21)( 13, 25)
( 14, 26)( 15, 27)( 16, 22)( 17, 23)( 18, 24)( 31, 34)( 32, 35)( 33, 36)
( 37, 46)( 38, 47)( 39, 48)( 40, 52)( 41, 53)( 42, 54)( 43, 49)( 44, 50)
( 45, 51)( 58, 61)( 59, 62)( 60, 63)( 64, 73)( 65, 74)( 66, 75)( 67, 79)
( 68, 80)( 69, 81)( 70, 76)( 71, 77)( 72, 78)( 85, 88)( 86, 89)( 87, 90)
( 91,100)( 92,101)( 93,102)( 94,106)( 95,107)( 96,108)( 97,103)( 98,104)
( 99,105)(112,115)(113,116)(114,117)(118,127)(119,128)(120,129)(121,133)
(122,134)(123,135)(124,130)(125,131)(126,132)(139,142)(140,143)(141,144)
(145,154)(146,155)(147,156)(148,160)(149,161)(150,162)(151,157)(152,158)
(153,159)(166,169)(167,170)(168,171)(172,181)(173,182)(174,183)(175,187)
(176,188)(177,189)(178,184)(179,185)(180,186)(193,196)(194,197)(195,198)
(199,208)(200,209)(201,210)(202,214)(203,215)(204,216)(205,211)(206,212)
(207,213);
s3 := Sym(216)!(  1, 10)(  2, 11)(  3, 12)(  4, 13)(  5, 14)(  6, 15)(  7, 16)
(  8, 17)(  9, 18)( 28, 37)( 29, 38)( 30, 39)( 31, 40)( 32, 41)( 33, 42)
( 34, 43)( 35, 44)( 36, 45)( 55, 64)( 56, 65)( 57, 66)( 58, 67)( 59, 68)
( 60, 69)( 61, 70)( 62, 71)( 63, 72)( 82, 91)( 83, 92)( 84, 93)( 85, 94)
( 86, 95)( 87, 96)( 88, 97)( 89, 98)( 90, 99)(109,118)(110,119)(111,120)
(112,121)(113,122)(114,123)(115,124)(116,125)(117,126)(136,145)(137,146)
(138,147)(139,148)(140,149)(141,150)(142,151)(143,152)(144,153)(163,172)
(164,173)(165,174)(166,175)(167,176)(168,177)(169,178)(170,179)(171,180)
(190,199)(191,200)(192,201)(193,202)(194,203)(195,204)(196,205)(197,206)
(198,207);
poly := sub<Sym(216)|s0,s1,s2,s3>;

Finitely Presented Group Representation (Magma) :

poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2, 
s3*s3, s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s1*s2*s3*s2*s1*s2*s3*s2, s2*s0*s1*s2*s1*s2*s0*s1*s2*s1, 
s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;

References : None.
to this polytope