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# Polytope of Type {4,18,6}

Atlas Canonical Name : {4,18,6}*864d
if this polytope has a name.
Group : SmallGroup(864,3999)
Rank : 4
Schlafli Type : {4,18,6}
Number of vertices, edges, etc : 4, 36, 54, 6
Order of s0s1s2s3 : 18
Order of s0s1s2s3s2s1 : 2
Special Properties :
Universal
Non-Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{4,18,6,2} of size 1728
Vertex Figure Of :
{2,4,18,6} of size 1728
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {4,9,6}*432
3-fold quotients : {4,18,2}*288b, {4,6,6}*288e
6-fold quotients : {4,9,2}*144, {4,3,6}*144
9-fold quotients : {4,6,2}*96c
18-fold quotients : {4,3,2}*48
Covers (Minimal Covers in Boldface) :
2-fold covers : {4,36,6}*1728d, {4,36,6}*1728f, {4,18,12}*1728d, {4,18,6}*1728b
Permutation Representation (GAP) :
```s0 := (  1,  3)(  2,  4)(  5,  7)(  6,  8)(  9, 11)( 10, 12)( 13, 15)( 14, 16)
( 17, 19)( 18, 20)( 21, 23)( 22, 24)( 25, 27)( 26, 28)( 29, 31)( 30, 32)
( 33, 35)( 34, 36)( 37, 39)( 38, 40)( 41, 43)( 42, 44)( 45, 47)( 46, 48)
( 49, 51)( 50, 52)( 53, 55)( 54, 56)( 57, 59)( 58, 60)( 61, 63)( 62, 64)
( 65, 67)( 66, 68)( 69, 71)( 70, 72)( 73, 75)( 74, 76)( 77, 79)( 78, 80)
( 81, 83)( 82, 84)( 85, 87)( 86, 88)( 89, 91)( 90, 92)( 93, 95)( 94, 96)
( 97, 99)( 98,100)(101,103)(102,104)(105,107)(106,108)(109,111)(110,112)
(113,115)(114,116)(117,119)(118,120)(121,123)(122,124)(125,127)(126,128)
(129,131)(130,132)(133,135)(134,136)(137,139)(138,140)(141,143)(142,144)
(145,147)(146,148)(149,151)(150,152)(153,155)(154,156)(157,159)(158,160)
(161,163)(162,164)(165,167)(166,168)(169,171)(170,172)(173,175)(174,176)
(177,179)(178,180)(181,183)(182,184)(185,187)(186,188)(189,191)(190,192)
(193,195)(194,196)(197,199)(198,200)(201,203)(202,204)(205,207)(206,208)
(209,211)(210,212)(213,215)(214,216);;
s1 := (  2,  3)(  5,  9)(  6, 11)(  7, 10)(  8, 12)( 13, 25)( 14, 27)( 15, 26)
( 16, 28)( 17, 33)( 18, 35)( 19, 34)( 20, 36)( 21, 29)( 22, 31)( 23, 30)
( 24, 32)( 37, 77)( 38, 79)( 39, 78)( 40, 80)( 41, 73)( 42, 75)( 43, 74)
( 44, 76)( 45, 81)( 46, 83)( 47, 82)( 48, 84)( 49,101)( 50,103)( 51,102)
( 52,104)( 53, 97)( 54, 99)( 55, 98)( 56,100)( 57,105)( 58,107)( 59,106)
( 60,108)( 61, 89)( 62, 91)( 63, 90)( 64, 92)( 65, 85)( 66, 87)( 67, 86)
( 68, 88)( 69, 93)( 70, 95)( 71, 94)( 72, 96)(110,111)(113,117)(114,119)
(115,118)(116,120)(121,133)(122,135)(123,134)(124,136)(125,141)(126,143)
(127,142)(128,144)(129,137)(130,139)(131,138)(132,140)(145,185)(146,187)
(147,186)(148,188)(149,181)(150,183)(151,182)(152,184)(153,189)(154,191)
(155,190)(156,192)(157,209)(158,211)(159,210)(160,212)(161,205)(162,207)
(163,206)(164,208)(165,213)(166,215)(167,214)(168,216)(169,197)(170,199)
(171,198)(172,200)(173,193)(174,195)(175,194)(176,196)(177,201)(178,203)
(179,202)(180,204);;
s2 := (  1,193)(  2,196)(  3,195)(  4,194)(  5,201)(  6,204)(  7,203)(  8,202)
(  9,197)( 10,200)( 11,199)( 12,198)( 13,181)( 14,184)( 15,183)( 16,182)
( 17,189)( 18,192)( 19,191)( 20,190)( 21,185)( 22,188)( 23,187)( 24,186)
( 25,205)( 26,208)( 27,207)( 28,206)( 29,213)( 30,216)( 31,215)( 32,214)
( 33,209)( 34,212)( 35,211)( 36,210)( 37,157)( 38,160)( 39,159)( 40,158)
( 41,165)( 42,168)( 43,167)( 44,166)( 45,161)( 46,164)( 47,163)( 48,162)
( 49,145)( 50,148)( 51,147)( 52,146)( 53,153)( 54,156)( 55,155)( 56,154)
( 57,149)( 58,152)( 59,151)( 60,150)( 61,169)( 62,172)( 63,171)( 64,170)
( 65,177)( 66,180)( 67,179)( 68,178)( 69,173)( 70,176)( 71,175)( 72,174)
( 73,121)( 74,124)( 75,123)( 76,122)( 77,129)( 78,132)( 79,131)( 80,130)
( 81,125)( 82,128)( 83,127)( 84,126)( 85,109)( 86,112)( 87,111)( 88,110)
( 89,117)( 90,120)( 91,119)( 92,118)( 93,113)( 94,116)( 95,115)( 96,114)
( 97,133)( 98,136)( 99,135)(100,134)(101,141)(102,144)(103,143)(104,142)
(105,137)(106,140)(107,139)(108,138);;
s3 := ( 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);;
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*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s2*s0*s1, s3*s1*s2*s3*s2*s3*s1*s2*s3*s2,
s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;

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

```
References : None.
to this polytope