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# Polytope of Type {36,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {36,4}*288c
if this polytope has a name.
Group : SmallGroup(288,334)
Rank : 3
Schlafli Type : {36,4}
Number of vertices, edges, etc : 36, 72, 4
Order of s0s1s2 : 36
Order of s0s1s2s1 : 4
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Non-Orientable
Flat
Self-Petrie
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Skewing Operation
Facet Of :
{36,4,2} of size 576
Vertex Figure Of :
{2,36,4} of size 576
{4,36,4} of size 1152
{4,36,4} of size 1152
{4,36,4} of size 1152
{6,36,4} of size 1728
{6,36,4} of size 1728
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {18,4}*144b
3-fold quotients : {12,4}*96c
4-fold quotients : {9,4}*72
6-fold quotients : {6,4}*48c
12-fold quotients : {3,4}*24
Covers (Minimal Covers in Boldface) :
2-fold covers : {36,4}*576b
3-fold covers : {108,4}*864c
4-fold covers : {36,4}*1152c, {36,4}*1152d, {36,8}*1152e, {36,8}*1152f, {72,4}*1152c, {72,4}*1152d
5-fold covers : {180,4}*1440c
6-fold covers : {108,4}*1728b, {36,12}*1728e, {36,12}*1728f
Permutation Representation (GAP) :
```s0 := (  2,  3)(  5,  9)(  6, 11)(  7, 10)(  8, 12)( 13, 29)( 14, 31)( 15, 30)
( 16, 32)( 17, 25)( 18, 27)( 19, 26)( 20, 28)( 21, 33)( 22, 35)( 23, 34)
( 24, 36)( 38, 39)( 41, 45)( 42, 47)( 43, 46)( 44, 48)( 49, 65)( 50, 67)
( 51, 66)( 52, 68)( 53, 61)( 54, 63)( 55, 62)( 56, 64)( 57, 69)( 58, 71)
( 59, 70)( 60, 72)( 73,109)( 74,111)( 75,110)( 76,112)( 77,117)( 78,119)
( 79,118)( 80,120)( 81,113)( 82,115)( 83,114)( 84,116)( 85,137)( 86,139)
( 87,138)( 88,140)( 89,133)( 90,135)( 91,134)( 92,136)( 93,141)( 94,143)
( 95,142)( 96,144)( 97,125)( 98,127)( 99,126)(100,128)(101,121)(102,123)
(103,122)(104,124)(105,129)(106,131)(107,130)(108,132);;
s1 := (  1, 85)(  2, 86)(  3, 88)(  4, 87)(  5, 93)(  6, 94)(  7, 96)(  8, 95)
(  9, 89)( 10, 90)( 11, 92)( 12, 91)( 13, 73)( 14, 74)( 15, 76)( 16, 75)
( 17, 81)( 18, 82)( 19, 84)( 20, 83)( 21, 77)( 22, 78)( 23, 80)( 24, 79)
( 25,101)( 26,102)( 27,104)( 28,103)( 29, 97)( 30, 98)( 31,100)( 32, 99)
( 33,105)( 34,106)( 35,108)( 36,107)( 37,121)( 38,122)( 39,124)( 40,123)
( 41,129)( 42,130)( 43,132)( 44,131)( 45,125)( 46,126)( 47,128)( 48,127)
( 49,109)( 50,110)( 51,112)( 52,111)( 53,117)( 54,118)( 55,120)( 56,119)
( 57,113)( 58,114)( 59,116)( 60,115)( 61,137)( 62,138)( 63,140)( 64,139)
( 65,133)( 66,134)( 67,136)( 68,135)( 69,141)( 70,142)( 71,144)( 72,143);;
s2 := (  1, 40)(  2, 39)(  3, 38)(  4, 37)(  5, 44)(  6, 43)(  7, 42)(  8, 41)
(  9, 48)( 10, 47)( 11, 46)( 12, 45)( 13, 52)( 14, 51)( 15, 50)( 16, 49)
( 17, 56)( 18, 55)( 19, 54)( 20, 53)( 21, 60)( 22, 59)( 23, 58)( 24, 57)
( 25, 64)( 26, 63)( 27, 62)( 28, 61)( 29, 68)( 30, 67)( 31, 66)( 32, 65)
( 33, 72)( 34, 71)( 35, 70)( 36, 69)( 73,112)( 74,111)( 75,110)( 76,109)
( 77,116)( 78,115)( 79,114)( 80,113)( 81,120)( 82,119)( 83,118)( 84,117)
( 85,124)( 86,123)( 87,122)( 88,121)( 89,128)( 90,127)( 91,126)( 92,125)
( 93,132)( 94,131)( 95,130)( 96,129)( 97,136)( 98,135)( 99,134)(100,133)
(101,140)(102,139)(103,138)(104,137)(105,144)(106,143)(107,142)(108,141);;
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, s1*s2*s1*s2*s1*s2*s1*s2,
s0*s1*s2*s1*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*s2*s1*s0*s2*s1*s0*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;

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

```
References : None.
to this polytope