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# Polytope of Type {2,24,6}

Atlas Canonical Name : {2,24,6}*1728c
if this polytope has a name.
Group : SmallGroup(1728,17452)
Rank : 4
Schlafli Type : {2,24,6}
Number of vertices, edges, etc : 2, 72, 216, 18
Order of s0s1s2s3 : 24
Order of s0s1s2s3s2s1 : 2
Special Properties :
Degenerate
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 : {2,12,6}*864c
3-fold quotients : {2,24,6}*576c
4-fold quotients : {2,6,6}*432a
6-fold quotients : {2,12,6}*288c
8-fold quotients : {2,6,3}*216
9-fold quotients : {2,8,6}*192
12-fold quotients : {2,6,6}*144b
18-fold quotients : {2,4,6}*96a
24-fold quotients : {2,6,3}*72
27-fold quotients : {2,8,2}*64
36-fold quotients : {2,2,6}*48
54-fold quotients : {2,4,2}*32
72-fold quotients : {2,2,3}*24
108-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := (1,2);;
s1 := (  4,  5)(  6,  9)(  7, 11)(  8, 10)( 13, 14)( 15, 18)( 16, 20)( 17, 19)
( 22, 23)( 24, 27)( 25, 29)( 26, 28)( 31, 32)( 33, 36)( 34, 38)( 35, 37)
( 40, 41)( 42, 45)( 43, 47)( 44, 46)( 49, 50)( 51, 54)( 52, 56)( 53, 55)
( 57, 84)( 58, 86)( 59, 85)( 60, 90)( 61, 92)( 62, 91)( 63, 87)( 64, 89)
( 65, 88)( 66, 93)( 67, 95)( 68, 94)( 69, 99)( 70,101)( 71,100)( 72, 96)
( 73, 98)( 74, 97)( 75,102)( 76,104)( 77,103)( 78,108)( 79,110)( 80,109)
( 81,105)( 82,107)( 83,106)(111,165)(112,167)(113,166)(114,171)(115,173)
(116,172)(117,168)(118,170)(119,169)(120,174)(121,176)(122,175)(123,180)
(124,182)(125,181)(126,177)(127,179)(128,178)(129,183)(130,185)(131,184)
(132,189)(133,191)(134,190)(135,186)(136,188)(137,187)(138,192)(139,194)
(140,193)(141,198)(142,200)(143,199)(144,195)(145,197)(146,196)(147,201)
(148,203)(149,202)(150,207)(151,209)(152,208)(153,204)(154,206)(155,205)
(156,210)(157,212)(158,211)(159,216)(160,218)(161,217)(162,213)(163,215)
(164,214);;
s2 := (  3,114)(  4,115)(  5,116)(  6,111)(  7,112)(  8,113)(  9,117)( 10,118)
( 11,119)( 12,132)( 13,133)( 14,134)( 15,129)( 16,130)( 17,131)( 18,135)
( 19,136)( 20,137)( 21,123)( 22,124)( 23,125)( 24,120)( 25,121)( 26,122)
( 27,126)( 28,127)( 29,128)( 30,141)( 31,142)( 32,143)( 33,138)( 34,139)
( 35,140)( 36,144)( 37,145)( 38,146)( 39,159)( 40,160)( 41,161)( 42,156)
( 43,157)( 44,158)( 45,162)( 46,163)( 47,164)( 48,150)( 49,151)( 50,152)
( 51,147)( 52,148)( 53,149)( 54,153)( 55,154)( 56,155)( 57,195)( 58,196)
( 59,197)( 60,192)( 61,193)( 62,194)( 63,198)( 64,199)( 65,200)( 66,213)
( 67,214)( 68,215)( 69,210)( 70,211)( 71,212)( 72,216)( 73,217)( 74,218)
( 75,204)( 76,205)( 77,206)( 78,201)( 79,202)( 80,203)( 81,207)( 82,208)
( 83,209)( 84,168)( 85,169)( 86,170)( 87,165)( 88,166)( 89,167)( 90,171)
( 91,172)( 92,173)( 93,186)( 94,187)( 95,188)( 96,183)( 97,184)( 98,185)
( 99,189)(100,190)(101,191)(102,177)(103,178)(104,179)(105,174)(106,175)
(107,176)(108,180)(109,181)(110,182);;
s3 := (  3, 12)(  4, 13)(  5, 14)(  6, 20)(  7, 18)(  8, 19)(  9, 16)( 10, 17)
( 11, 15)( 24, 29)( 25, 27)( 26, 28)( 30, 39)( 31, 40)( 32, 41)( 33, 47)
( 34, 45)( 35, 46)( 36, 43)( 37, 44)( 38, 42)( 51, 56)( 52, 54)( 53, 55)
( 57, 66)( 58, 67)( 59, 68)( 60, 74)( 61, 72)( 62, 73)( 63, 70)( 64, 71)
( 65, 69)( 78, 83)( 79, 81)( 80, 82)( 84, 93)( 85, 94)( 86, 95)( 87,101)
( 88, 99)( 89,100)( 90, 97)( 91, 98)( 92, 96)(105,110)(106,108)(107,109)
(111,120)(112,121)(113,122)(114,128)(115,126)(116,127)(117,124)(118,125)
(119,123)(132,137)(133,135)(134,136)(138,147)(139,148)(140,149)(141,155)
(142,153)(143,154)(144,151)(145,152)(146,150)(159,164)(160,162)(161,163)
(165,174)(166,175)(167,176)(168,182)(169,180)(170,181)(171,178)(172,179)
(173,177)(186,191)(187,189)(188,190)(192,201)(193,202)(194,203)(195,209)
(196,207)(197,208)(198,205)(199,206)(200,204)(213,218)(214,216)(215,217);;
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*s1*s0*s1,
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
s3*s1*s2*s1*s2*s3*s2*s1*s2*s3*s1*s2*s1*s2*s3*s2*s1*s2,
s2*s3*s1*s2*s1*s3*s2*s1*s2*s1*s3*s2*s1*s2*s1*s2*s3*s1*s2*s1 ];;
poly := F / rels;;

Permutation Representation (Magma) :
s0 := Sym(218)!(1,2);
s1 := Sym(218)!(  4,  5)(  6,  9)(  7, 11)(  8, 10)( 13, 14)( 15, 18)( 16, 20)
( 17, 19)( 22, 23)( 24, 27)( 25, 29)( 26, 28)( 31, 32)( 33, 36)( 34, 38)
( 35, 37)( 40, 41)( 42, 45)( 43, 47)( 44, 46)( 49, 50)( 51, 54)( 52, 56)
( 53, 55)( 57, 84)( 58, 86)( 59, 85)( 60, 90)( 61, 92)( 62, 91)( 63, 87)
( 64, 89)( 65, 88)( 66, 93)( 67, 95)( 68, 94)( 69, 99)( 70,101)( 71,100)
( 72, 96)( 73, 98)( 74, 97)( 75,102)( 76,104)( 77,103)( 78,108)( 79,110)
( 80,109)( 81,105)( 82,107)( 83,106)(111,165)(112,167)(113,166)(114,171)
(115,173)(116,172)(117,168)(118,170)(119,169)(120,174)(121,176)(122,175)
(123,180)(124,182)(125,181)(126,177)(127,179)(128,178)(129,183)(130,185)
(131,184)(132,189)(133,191)(134,190)(135,186)(136,188)(137,187)(138,192)
(139,194)(140,193)(141,198)(142,200)(143,199)(144,195)(145,197)(146,196)
(147,201)(148,203)(149,202)(150,207)(151,209)(152,208)(153,204)(154,206)
(155,205)(156,210)(157,212)(158,211)(159,216)(160,218)(161,217)(162,213)
(163,215)(164,214);
s2 := Sym(218)!(  3,114)(  4,115)(  5,116)(  6,111)(  7,112)(  8,113)(  9,117)
( 10,118)( 11,119)( 12,132)( 13,133)( 14,134)( 15,129)( 16,130)( 17,131)
( 18,135)( 19,136)( 20,137)( 21,123)( 22,124)( 23,125)( 24,120)( 25,121)
( 26,122)( 27,126)( 28,127)( 29,128)( 30,141)( 31,142)( 32,143)( 33,138)
( 34,139)( 35,140)( 36,144)( 37,145)( 38,146)( 39,159)( 40,160)( 41,161)
( 42,156)( 43,157)( 44,158)( 45,162)( 46,163)( 47,164)( 48,150)( 49,151)
( 50,152)( 51,147)( 52,148)( 53,149)( 54,153)( 55,154)( 56,155)( 57,195)
( 58,196)( 59,197)( 60,192)( 61,193)( 62,194)( 63,198)( 64,199)( 65,200)
( 66,213)( 67,214)( 68,215)( 69,210)( 70,211)( 71,212)( 72,216)( 73,217)
( 74,218)( 75,204)( 76,205)( 77,206)( 78,201)( 79,202)( 80,203)( 81,207)
( 82,208)( 83,209)( 84,168)( 85,169)( 86,170)( 87,165)( 88,166)( 89,167)
( 90,171)( 91,172)( 92,173)( 93,186)( 94,187)( 95,188)( 96,183)( 97,184)
( 98,185)( 99,189)(100,190)(101,191)(102,177)(103,178)(104,179)(105,174)
(106,175)(107,176)(108,180)(109,181)(110,182);
s3 := Sym(218)!(  3, 12)(  4, 13)(  5, 14)(  6, 20)(  7, 18)(  8, 19)(  9, 16)
( 10, 17)( 11, 15)( 24, 29)( 25, 27)( 26, 28)( 30, 39)( 31, 40)( 32, 41)
( 33, 47)( 34, 45)( 35, 46)( 36, 43)( 37, 44)( 38, 42)( 51, 56)( 52, 54)
( 53, 55)( 57, 66)( 58, 67)( 59, 68)( 60, 74)( 61, 72)( 62, 73)( 63, 70)
( 64, 71)( 65, 69)( 78, 83)( 79, 81)( 80, 82)( 84, 93)( 85, 94)( 86, 95)
( 87,101)( 88, 99)( 89,100)( 90, 97)( 91, 98)( 92, 96)(105,110)(106,108)
(107,109)(111,120)(112,121)(113,122)(114,128)(115,126)(116,127)(117,124)
(118,125)(119,123)(132,137)(133,135)(134,136)(138,147)(139,148)(140,149)
(141,155)(142,153)(143,154)(144,151)(145,152)(146,150)(159,164)(160,162)
(161,163)(165,174)(166,175)(167,176)(168,182)(169,180)(170,181)(171,178)
(172,179)(173,177)(186,191)(187,189)(188,190)(192,201)(193,202)(194,203)
(195,209)(196,207)(197,208)(198,205)(199,206)(200,204)(213,218)(214,216)
(215,217);
poly := sub<Sym(218)|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*s1*s0*s1, s0*s2*s0*s2, s0*s3*s0*s3,
s1*s3*s1*s3, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
s3*s1*s2*s1*s2*s3*s2*s1*s2*s3*s1*s2*s1*s2*s3*s2*s1*s2,
s2*s3*s1*s2*s1*s3*s2*s1*s2*s1*s3*s2*s1*s2*s1*s2*s3*s1*s2*s1 >;

to this polytope