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# Polytope of Type {18,8}

Atlas Canonical Name : {18,8}*576c
if this polytope has a name.
Group : SmallGroup(576,4983)
Rank : 3
Schlafli Type : {18,8}
Number of vertices, edges, etc : 36, 144, 16
Order of s0s1s2 : 18
Order of s0s1s2s1 : 8
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Self-Petrie
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Facet Of :
{18,8,2} of size 1152
Vertex Figure Of :
{2,18,8} of size 1152
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {18,4}*288
3-fold quotients : {6,8}*192c
4-fold quotients : {9,4}*144, {18,4}*144b, {18,4}*144c
6-fold quotients : {6,4}*96
8-fold quotients : {9,4}*72, {18,2}*72
12-fold quotients : {3,4}*48, {6,4}*48b, {6,4}*48c
16-fold quotients : {9,2}*36
24-fold quotients : {3,4}*24, {6,2}*24
48-fold quotients : {3,2}*12
72-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {36,8}*1152f, {18,8}*1152f, {36,8}*1152g
3-fold covers : {54,8}*1728c, {18,24}*1728c, {18,24}*1728d
Permutation Representation (GAP) :
```s0 := (  3,  6)(  4,  5)(  7,  8)(  9, 17)( 10, 18)( 11, 22)( 12, 21)( 13, 20)
( 14, 19)( 15, 24)( 16, 23)( 25, 65)( 26, 66)( 27, 70)( 28, 69)( 29, 68)
( 30, 67)( 31, 72)( 32, 71)( 33, 57)( 34, 58)( 35, 62)( 36, 61)( 37, 60)
( 38, 59)( 39, 64)( 40, 63)( 41, 49)( 42, 50)( 43, 54)( 44, 53)( 45, 52)
( 46, 51)( 47, 56)( 48, 55)( 73, 74)( 75, 77)( 76, 78)( 81, 90)( 82, 89)
( 83, 93)( 84, 94)( 85, 91)( 86, 92)( 87, 95)( 88, 96)( 97,138)( 98,137)
( 99,141)(100,142)(101,139)(102,140)(103,143)(104,144)(105,130)(106,129)
(107,133)(108,134)(109,131)(110,132)(111,135)(112,136)(113,122)(114,121)
(115,125)(116,126)(117,123)(118,124)(119,127)(120,128);;
s1 := (  1, 25)(  2, 26)(  3, 28)(  4, 27)(  5, 31)(  6, 32)(  7, 29)(  8, 30)
(  9, 41)( 10, 42)( 11, 44)( 12, 43)( 13, 47)( 14, 48)( 15, 45)( 16, 46)
( 17, 33)( 18, 34)( 19, 36)( 20, 35)( 21, 39)( 22, 40)( 23, 37)( 24, 38)
( 49, 65)( 50, 66)( 51, 68)( 52, 67)( 53, 71)( 54, 72)( 55, 69)( 56, 70)
( 59, 60)( 61, 63)( 62, 64)( 73, 97)( 74, 98)( 75,100)( 76, 99)( 77,103)
( 78,104)( 79,101)( 80,102)( 81,113)( 82,114)( 83,116)( 84,115)( 85,119)
( 86,120)( 87,117)( 88,118)( 89,105)( 90,106)( 91,108)( 92,107)( 93,111)
( 94,112)( 95,109)( 96,110)(121,137)(122,138)(123,140)(124,139)(125,143)
(126,144)(127,141)(128,142)(131,132)(133,135)(134,136);;
s2 := (  1, 79)(  2, 80)(  3, 77)(  4, 78)(  5, 76)(  6, 75)(  7, 74)(  8, 73)
(  9, 87)( 10, 88)( 11, 85)( 12, 86)( 13, 84)( 14, 83)( 15, 82)( 16, 81)
( 17, 95)( 18, 96)( 19, 93)( 20, 94)( 21, 92)( 22, 91)( 23, 90)( 24, 89)
( 25,103)( 26,104)( 27,101)( 28,102)( 29,100)( 30, 99)( 31, 98)( 32, 97)
( 33,111)( 34,112)( 35,109)( 36,110)( 37,108)( 38,107)( 39,106)( 40,105)
( 41,119)( 42,120)( 43,117)( 44,118)( 45,116)( 46,115)( 47,114)( 48,113)
( 49,127)( 50,128)( 51,125)( 52,126)( 53,124)( 54,123)( 55,122)( 56,121)
( 57,135)( 58,136)( 59,133)( 60,134)( 61,132)( 62,131)( 63,130)( 64,129)
( 65,143)( 66,144)( 67,141)( 68,142)( 69,140)( 70,139)( 71,138)( 72,137);;
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, s2*s0*s1*s2*s1*s2*s0*s1*s0*s1*s2*s1*s0*s1,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;

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

```
References : None.
to this polytope