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# Polytope of Type {6,4,9}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,4,9}*864
if this polytope has a name.
Group : SmallGroup(864,3999)
Rank : 4
Schlafli Type : {6,4,9}
Number of vertices, edges, etc : 6, 24, 36, 18
Order of s0s1s2s3 : 18
Order of s0s1s2s3s2s1 : 2
Special Properties :
Universal
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{6,4,9,2} of size 1728
Vertex Figure Of :
{2,6,4,9} of size 1728
Quotients (Maximal Quotients in Boldface) :
3-fold quotients : {2,4,9}*288, {6,4,3}*288
4-fold quotients : {6,2,9}*216
6-fold quotients : {2,4,9}*144
8-fold quotients : {3,2,9}*108
9-fold quotients : {2,4,3}*96
12-fold quotients : {2,2,9}*72, {6,2,3}*72
18-fold quotients : {2,4,3}*48
24-fold quotients : {3,2,3}*36
36-fold quotients : {2,2,3}*24
Covers (Minimal Covers in Boldface) :
2-fold covers : {12,4,9}*1728, {6,8,9}*1728, {6,4,18}*1728b
Permutation Representation (GAP) :
```s0 := ( 13, 25)( 14, 26)( 15, 27)( 16, 28)( 17, 29)( 18, 30)( 19, 31)( 20, 32)
( 21, 33)( 22, 34)( 23, 35)( 24, 36)( 49, 61)( 50, 62)( 51, 63)( 52, 64)
( 53, 65)( 54, 66)( 55, 67)( 56, 68)( 57, 69)( 58, 70)( 59, 71)( 60, 72)
( 85, 97)( 86, 98)( 87, 99)( 88,100)( 89,101)( 90,102)( 91,103)( 92,104)
( 93,105)( 94,106)( 95,107)( 96,108)(121,133)(122,134)(123,135)(124,136)
(125,137)(126,138)(127,139)(128,140)(129,141)(130,142)(131,143)(132,144)
(157,169)(158,170)(159,171)(160,172)(161,173)(162,174)(163,175)(164,176)
(165,177)(166,178)(167,179)(168,180)(193,205)(194,206)(195,207)(196,208)
(197,209)(198,210)(199,211)(200,212)(201,213)(202,214)(203,215)(204,216);;
s1 := (  1,123)(  2,124)(  3,121)(  4,122)(  5,127)(  6,128)(  7,125)(  8,126)
(  9,131)( 10,132)( 11,129)( 12,130)( 13,111)( 14,112)( 15,109)( 16,110)
( 17,115)( 18,116)( 19,113)( 20,114)( 21,119)( 22,120)( 23,117)( 24,118)
( 25,135)( 26,136)( 27,133)( 28,134)( 29,139)( 30,140)( 31,137)( 32,138)
( 33,143)( 34,144)( 35,141)( 36,142)( 37,159)( 38,160)( 39,157)( 40,158)
( 41,163)( 42,164)( 43,161)( 44,162)( 45,167)( 46,168)( 47,165)( 48,166)
( 49,147)( 50,148)( 51,145)( 52,146)( 53,151)( 54,152)( 55,149)( 56,150)
( 57,155)( 58,156)( 59,153)( 60,154)( 61,171)( 62,172)( 63,169)( 64,170)
( 65,175)( 66,176)( 67,173)( 68,174)( 69,179)( 70,180)( 71,177)( 72,178)
( 73,195)( 74,196)( 75,193)( 76,194)( 77,199)( 78,200)( 79,197)( 80,198)
( 81,203)( 82,204)( 83,201)( 84,202)( 85,183)( 86,184)( 87,181)( 88,182)
( 89,187)( 90,188)( 91,185)( 92,186)( 93,191)( 94,192)( 95,189)( 96,190)
( 97,207)( 98,208)( 99,205)(100,206)(101,211)(102,212)(103,209)(104,210)
(105,215)(106,216)(107,213)(108,214);;
s2 := (  2,  3)(  5,  9)(  6, 11)(  7, 10)(  8, 12)( 14, 15)( 17, 21)( 18, 23)
( 19, 22)( 20, 24)( 26, 27)( 29, 33)( 30, 35)( 31, 34)( 32, 36)( 37, 77)
( 38, 79)( 39, 78)( 40, 80)( 41, 73)( 42, 75)( 43, 74)( 44, 76)( 45, 81)
( 46, 83)( 47, 82)( 48, 84)( 49, 89)( 50, 91)( 51, 90)( 52, 92)( 53, 85)
( 54, 87)( 55, 86)( 56, 88)( 57, 93)( 58, 95)( 59, 94)( 60, 96)( 61,101)
( 62,103)( 63,102)( 64,104)( 65, 97)( 66, 99)( 67, 98)( 68,100)( 69,105)
( 70,107)( 71,106)( 72,108)(110,111)(113,117)(114,119)(115,118)(116,120)
(122,123)(125,129)(126,131)(127,130)(128,132)(134,135)(137,141)(138,143)
(139,142)(140,144)(145,185)(146,187)(147,186)(148,188)(149,181)(150,183)
(151,182)(152,184)(153,189)(154,191)(155,190)(156,192)(157,197)(158,199)
(159,198)(160,200)(161,193)(162,195)(163,194)(164,196)(165,201)(166,203)
(167,202)(168,204)(169,209)(170,211)(171,210)(172,212)(173,205)(174,207)
(175,206)(176,208)(177,213)(178,215)(179,214)(180,216);;
s3 := (  1, 73)(  2, 76)(  3, 75)(  4, 74)(  5, 81)(  6, 84)(  7, 83)(  8, 82)
(  9, 77)( 10, 80)( 11, 79)( 12, 78)( 13, 85)( 14, 88)( 15, 87)( 16, 86)
( 17, 93)( 18, 96)( 19, 95)( 20, 94)( 21, 89)( 22, 92)( 23, 91)( 24, 90)
( 25, 97)( 26,100)( 27, 99)( 28, 98)( 29,105)( 30,108)( 31,107)( 32,106)
( 33,101)( 34,104)( 35,103)( 36,102)( 38, 40)( 41, 45)( 42, 48)( 43, 47)
( 44, 46)( 50, 52)( 53, 57)( 54, 60)( 55, 59)( 56, 58)( 62, 64)( 65, 69)
( 66, 72)( 67, 71)( 68, 70)(109,181)(110,184)(111,183)(112,182)(113,189)
(114,192)(115,191)(116,190)(117,185)(118,188)(119,187)(120,186)(121,193)
(122,196)(123,195)(124,194)(125,201)(126,204)(127,203)(128,202)(129,197)
(130,200)(131,199)(132,198)(133,205)(134,208)(135,207)(136,206)(137,213)
(138,216)(139,215)(140,214)(141,209)(142,212)(143,211)(144,210)(146,148)
(149,153)(150,156)(151,155)(152,154)(158,160)(161,165)(162,168)(163,167)
(164,166)(170,172)(173,177)(174,180)(175,179)(176,178);;
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, s0*s1*s2*s1*s0*s1*s2*s1,
s1*s2*s1*s2*s1*s2*s1*s2, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s1*s2*s3*s2*s3*s2*s1*s2*s3*s2*s3*s2,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;

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

```
References : None.
to this polytope