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# Polytope of Type {2,18,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,18,6}*1296h
if this polytope has a name.
Group : SmallGroup(1296,1862)
Rank : 4
Schlafli Type : {2,18,6}
Number of vertices, edges, etc : 2, 54, 162, 18
Order of s0s1s2s3 : 6
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 : {2,18,3}*648
3-fold quotients : {2,6,6}*432a
6-fold quotients : {2,6,3}*216
9-fold quotients : {2,6,6}*144b
18-fold quotients : {2,6,3}*72
27-fold quotients : {2,2,6}*48
54-fold quotients : {2,2,3}*24
81-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)( 30, 59)( 31, 58)( 32, 57)( 33, 65)
( 34, 64)( 35, 63)( 36, 62)( 37, 61)( 38, 60)( 39, 68)( 40, 67)( 41, 66)
( 42, 74)( 43, 73)( 44, 72)( 45, 71)( 46, 70)( 47, 69)( 48, 77)( 49, 76)
( 50, 75)( 51, 83)( 52, 82)( 53, 81)( 54, 80)( 55, 79)( 56, 78)( 85, 86)
( 87, 90)( 88, 92)( 89, 91)( 94, 95)( 96, 99)( 97,101)( 98,100)(103,104)
(105,108)(106,110)(107,109)(111,140)(112,139)(113,138)(114,146)(115,145)
(116,144)(117,143)(118,142)(119,141)(120,149)(121,148)(122,147)(123,155)
(124,154)(125,153)(126,152)(127,151)(128,150)(129,158)(130,157)(131,156)
(132,164)(133,163)(134,162)(135,161)(136,160)(137,159);;
s2 := (  3, 30)(  4, 32)(  5, 31)(  6, 35)(  7, 34)(  8, 33)(  9, 37)( 10, 36)
( 11, 38)( 12, 55)( 13, 54)( 14, 56)( 15, 48)( 16, 50)( 17, 49)( 18, 53)
( 19, 52)( 20, 51)( 21, 42)( 22, 44)( 23, 43)( 24, 47)( 25, 46)( 26, 45)
( 27, 40)( 28, 39)( 29, 41)( 57, 59)( 60, 61)( 64, 65)( 66, 81)( 67, 83)
( 68, 82)( 69, 77)( 70, 76)( 71, 75)( 72, 79)( 73, 78)( 74, 80)( 84,111)
( 85,113)( 86,112)( 87,116)( 88,115)( 89,114)( 90,118)( 91,117)( 92,119)
( 93,136)( 94,135)( 95,137)( 96,129)( 97,131)( 98,130)( 99,134)(100,133)
(101,132)(102,123)(103,125)(104,124)(105,128)(106,127)(107,126)(108,121)
(109,120)(110,122)(138,140)(141,142)(145,146)(147,162)(148,164)(149,163)
(150,158)(151,157)(152,156)(153,160)(154,159)(155,161);;
s3 := (  3, 93)(  4, 95)(  5, 94)(  6, 96)(  7, 98)(  8, 97)(  9, 99)( 10,101)
( 11,100)( 12, 84)( 13, 86)( 14, 85)( 15, 87)( 16, 89)( 17, 88)( 18, 90)
( 19, 92)( 20, 91)( 21,102)( 22,104)( 23,103)( 24,105)( 25,107)( 26,106)
( 27,108)( 28,110)( 29,109)( 30,149)( 31,148)( 32,147)( 33,152)( 34,151)
( 35,150)( 36,155)( 37,154)( 38,153)( 39,140)( 40,139)( 41,138)( 42,143)
( 43,142)( 44,141)( 45,146)( 46,145)( 47,144)( 48,158)( 49,157)( 50,156)
( 51,161)( 52,160)( 53,159)( 54,164)( 55,163)( 56,162)( 57,122)( 58,121)
( 59,120)( 60,125)( 61,124)( 62,123)( 63,128)( 64,127)( 65,126)( 66,113)
( 67,112)( 68,111)( 69,116)( 70,115)( 71,114)( 72,119)( 73,118)( 74,117)
( 75,131)( 76,130)( 77,129)( 78,134)( 79,133)( 80,132)( 81,137)( 82,136)
( 83,135);;
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,
s1*s2*s3*s2*s3*s2*s1*s2*s3*s2*s3*s2,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2*s3*s1*s2*s1*s3*s2,
s3*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s3*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := Sym(164)!(1,2);
s1 := Sym(164)!(  4,  5)(  6,  9)(  7, 11)(  8, 10)( 13, 14)( 15, 18)( 16, 20)
( 17, 19)( 22, 23)( 24, 27)( 25, 29)( 26, 28)( 30, 59)( 31, 58)( 32, 57)
( 33, 65)( 34, 64)( 35, 63)( 36, 62)( 37, 61)( 38, 60)( 39, 68)( 40, 67)
( 41, 66)( 42, 74)( 43, 73)( 44, 72)( 45, 71)( 46, 70)( 47, 69)( 48, 77)
( 49, 76)( 50, 75)( 51, 83)( 52, 82)( 53, 81)( 54, 80)( 55, 79)( 56, 78)
( 85, 86)( 87, 90)( 88, 92)( 89, 91)( 94, 95)( 96, 99)( 97,101)( 98,100)
(103,104)(105,108)(106,110)(107,109)(111,140)(112,139)(113,138)(114,146)
(115,145)(116,144)(117,143)(118,142)(119,141)(120,149)(121,148)(122,147)
(123,155)(124,154)(125,153)(126,152)(127,151)(128,150)(129,158)(130,157)
(131,156)(132,164)(133,163)(134,162)(135,161)(136,160)(137,159);
s2 := Sym(164)!(  3, 30)(  4, 32)(  5, 31)(  6, 35)(  7, 34)(  8, 33)(  9, 37)
( 10, 36)( 11, 38)( 12, 55)( 13, 54)( 14, 56)( 15, 48)( 16, 50)( 17, 49)
( 18, 53)( 19, 52)( 20, 51)( 21, 42)( 22, 44)( 23, 43)( 24, 47)( 25, 46)
( 26, 45)( 27, 40)( 28, 39)( 29, 41)( 57, 59)( 60, 61)( 64, 65)( 66, 81)
( 67, 83)( 68, 82)( 69, 77)( 70, 76)( 71, 75)( 72, 79)( 73, 78)( 74, 80)
( 84,111)( 85,113)( 86,112)( 87,116)( 88,115)( 89,114)( 90,118)( 91,117)
( 92,119)( 93,136)( 94,135)( 95,137)( 96,129)( 97,131)( 98,130)( 99,134)
(100,133)(101,132)(102,123)(103,125)(104,124)(105,128)(106,127)(107,126)
(108,121)(109,120)(110,122)(138,140)(141,142)(145,146)(147,162)(148,164)
(149,163)(150,158)(151,157)(152,156)(153,160)(154,159)(155,161);
s3 := Sym(164)!(  3, 93)(  4, 95)(  5, 94)(  6, 96)(  7, 98)(  8, 97)(  9, 99)
( 10,101)( 11,100)( 12, 84)( 13, 86)( 14, 85)( 15, 87)( 16, 89)( 17, 88)
( 18, 90)( 19, 92)( 20, 91)( 21,102)( 22,104)( 23,103)( 24,105)( 25,107)
( 26,106)( 27,108)( 28,110)( 29,109)( 30,149)( 31,148)( 32,147)( 33,152)
( 34,151)( 35,150)( 36,155)( 37,154)( 38,153)( 39,140)( 40,139)( 41,138)
( 42,143)( 43,142)( 44,141)( 45,146)( 46,145)( 47,144)( 48,158)( 49,157)
( 50,156)( 51,161)( 52,160)( 53,159)( 54,164)( 55,163)( 56,162)( 57,122)
( 58,121)( 59,120)( 60,125)( 61,124)( 62,123)( 63,128)( 64,127)( 65,126)
( 66,113)( 67,112)( 68,111)( 69,116)( 70,115)( 71,114)( 72,119)( 73,118)
( 74,117)( 75,131)( 76,130)( 77,129)( 78,134)( 79,133)( 80,132)( 81,137)
( 82,136)( 83,135);
poly := sub<Sym(164)|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, s1*s2*s3*s2*s3*s2*s1*s2*s3*s2*s3*s2,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2*s3*s1*s2*s1*s3*s2,
s3*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s3*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;

```

to this polytope