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# Polytope of Type {6,36}

Atlas Canonical Name : {6,36}*1296m
if this polytope has a name.
Group : SmallGroup(1296,2976)
Rank : 3
Schlafli Type : {6,36}
Number of vertices, edges, etc : 18, 324, 108
Order of s0s1s2 : 36
Order of s0s1s2s1 : 6
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
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 : {6,36}*648a
3-fold quotients : {6,12}*432i
6-fold quotients : {6,12}*216c
9-fold quotients : {2,36}*144, {6,4}*144
18-fold quotients : {2,18}*72, {6,4}*72
27-fold quotients : {2,12}*48
36-fold quotients : {2,9}*36
54-fold quotients : {2,6}*24
81-fold quotients : {2,4}*16
108-fold quotients : {2,3}*12
162-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
```s0 := (  1, 82)(  2, 83)(  3, 84)(  4, 85)(  5, 86)(  6, 87)(  7, 88)(  8, 89)
(  9, 90)( 10,100)( 11,101)( 12,102)( 13,103)( 14,104)( 15,105)( 16,106)
( 17,107)( 18,108)( 19, 91)( 20, 92)( 21, 93)( 22, 94)( 23, 95)( 24, 96)
( 25, 97)( 26, 98)( 27, 99)( 28,136)( 29,137)( 30,138)( 31,139)( 32,140)
( 33,141)( 34,142)( 35,143)( 36,144)( 37,154)( 38,155)( 39,156)( 40,157)
( 41,158)( 42,159)( 43,160)( 44,161)( 45,162)( 46,145)( 47,146)( 48,147)
( 49,148)( 50,149)( 51,150)( 52,151)( 53,152)( 54,153)( 55,109)( 56,110)
( 57,111)( 58,112)( 59,113)( 60,114)( 61,115)( 62,116)( 63,117)( 64,127)
( 65,128)( 66,129)( 67,130)( 68,131)( 69,132)( 70,133)( 71,134)( 72,135)
( 73,118)( 74,119)( 75,120)( 76,121)( 77,122)( 78,123)( 79,124)( 80,125)
( 81,126);;
s1 := (  1, 28)(  2, 30)(  3, 29)(  4, 36)(  5, 35)(  6, 34)(  7, 33)(  8, 32)
(  9, 31)( 10, 37)( 11, 39)( 12, 38)( 13, 45)( 14, 44)( 15, 43)( 16, 42)
( 17, 41)( 18, 40)( 19, 46)( 20, 48)( 21, 47)( 22, 54)( 23, 53)( 24, 52)
( 25, 51)( 26, 50)( 27, 49)( 56, 57)( 58, 63)( 59, 62)( 60, 61)( 65, 66)
( 67, 72)( 68, 71)( 69, 70)( 74, 75)( 76, 81)( 77, 80)( 78, 79)( 82,109)
( 83,111)( 84,110)( 85,117)( 86,116)( 87,115)( 88,114)( 89,113)( 90,112)
( 91,118)( 92,120)( 93,119)( 94,126)( 95,125)( 96,124)( 97,123)( 98,122)
( 99,121)(100,127)(101,129)(102,128)(103,135)(104,134)(105,133)(106,132)
(107,131)(108,130)(137,138)(139,144)(140,143)(141,142)(146,147)(148,153)
(149,152)(150,151)(155,156)(157,162)(158,161)(159,160);;
s2 := (  1,  4)(  2,  6)(  3,  5)(  7,  9)( 10, 58)( 11, 60)( 12, 59)( 13, 55)
( 14, 57)( 15, 56)( 16, 63)( 17, 62)( 18, 61)( 19, 31)( 20, 33)( 21, 32)
( 22, 28)( 23, 30)( 24, 29)( 25, 36)( 26, 35)( 27, 34)( 37, 76)( 38, 78)
( 39, 77)( 40, 73)( 41, 75)( 42, 74)( 43, 81)( 44, 80)( 45, 79)( 46, 49)
( 47, 51)( 48, 50)( 52, 54)( 64, 67)( 65, 69)( 66, 68)( 70, 72)( 82, 85)
( 83, 87)( 84, 86)( 88, 90)( 91,139)( 92,141)( 93,140)( 94,136)( 95,138)
( 96,137)( 97,144)( 98,143)( 99,142)(100,112)(101,114)(102,113)(103,109)
(104,111)(105,110)(106,117)(107,116)(108,115)(118,157)(119,159)(120,158)
(121,154)(122,156)(123,155)(124,162)(125,161)(126,160)(127,130)(128,132)
(129,131)(133,135)(145,148)(146,150)(147,149)(151,153);;
poly := Group([s0,s1,s2]);;

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

```
Finitely Presented Group Representation (Magma) :
```poly<s0,s1,s2> := Group< s0,s1,s2 | s0*s0, s1*s1, s2*s2,
s0*s2*s0*s2, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s2*s0*s1*s2*s0*s1*s0*s1*s2*s1*s2*s1*s0*s1,
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*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