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# Polytope of Type {4,72}

Atlas Canonical Name : {4,72}*576a
Also Known As : {4,72|2}. if this polytope has another name.
Group : SmallGroup(576,296)
Rank : 3
Schlafli Type : {4,72}
Number of vertices, edges, etc : 4, 144, 72
Order of s0s1s2 : 72
Order of s0s1s2s1 : 2
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{4,72,2} of size 1152
Vertex Figure Of :
{2,4,72} of size 1152
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {4,36}*288a, {2,72}*288
3-fold quotients : {4,24}*192a
4-fold quotients : {2,36}*144, {4,18}*144a
6-fold quotients : {4,12}*96a, {2,24}*96
8-fold quotients : {2,18}*72
9-fold quotients : {4,8}*64a
12-fold quotients : {2,12}*48, {4,6}*48a
16-fold quotients : {2,9}*36
18-fold quotients : {4,4}*32, {2,8}*32
24-fold quotients : {2,6}*24
36-fold quotients : {2,4}*16, {4,2}*16
48-fold quotients : {2,3}*12
72-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {4,72}*1152a, {8,72}*1152b, {8,72}*1152c, {4,144}*1152a, {4,144}*1152b
3-fold covers : {4,216}*1728a, {12,72}*1728a, {12,72}*1728b
Permutation Representation (GAP) :
```s0 := ( 73, 91)( 74, 92)( 75, 93)( 76, 94)( 77, 95)( 78, 96)( 79, 97)( 80, 98)
( 81, 99)( 82,100)( 83,101)( 84,102)( 85,103)( 86,104)( 87,105)( 88,106)
( 89,107)( 90,108)(109,127)(110,128)(111,129)(112,130)(113,131)(114,132)
(115,133)(116,134)(117,135)(118,136)(119,137)(120,138)(121,139)(122,140)
(123,141)(124,142)(125,143)(126,144);;
s1 := (  1, 73)(  2, 75)(  3, 74)(  4, 80)(  5, 79)(  6, 81)(  7, 77)(  8, 76)
(  9, 78)( 10, 82)( 11, 84)( 12, 83)( 13, 89)( 14, 88)( 15, 90)( 16, 86)
( 17, 85)( 18, 87)( 19, 91)( 20, 93)( 21, 92)( 22, 98)( 23, 97)( 24, 99)
( 25, 95)( 26, 94)( 27, 96)( 28,100)( 29,102)( 30,101)( 31,107)( 32,106)
( 33,108)( 34,104)( 35,103)( 36,105)( 37,118)( 38,120)( 39,119)( 40,125)
( 41,124)( 42,126)( 43,122)( 44,121)( 45,123)( 46,109)( 47,111)( 48,110)
( 49,116)( 50,115)( 51,117)( 52,113)( 53,112)( 54,114)( 55,136)( 56,138)
( 57,137)( 58,143)( 59,142)( 60,144)( 61,140)( 62,139)( 63,141)( 64,127)
( 65,129)( 66,128)( 67,134)( 68,133)( 69,135)( 70,131)( 71,130)( 72,132);;
s2 := (  1,  4)(  2,  6)(  3,  5)(  7,  8)( 10, 13)( 11, 15)( 12, 14)( 16, 17)
( 19, 22)( 20, 24)( 21, 23)( 25, 26)( 28, 31)( 29, 33)( 30, 32)( 34, 35)
( 37, 49)( 38, 51)( 39, 50)( 40, 46)( 41, 48)( 42, 47)( 43, 53)( 44, 52)
( 45, 54)( 55, 67)( 56, 69)( 57, 68)( 58, 64)( 59, 66)( 60, 65)( 61, 71)
( 62, 70)( 63, 72)( 73,112)( 74,114)( 75,113)( 76,109)( 77,111)( 78,110)
( 79,116)( 80,115)( 81,117)( 82,121)( 83,123)( 84,122)( 85,118)( 86,120)
( 87,119)( 88,125)( 89,124)( 90,126)( 91,130)( 92,132)( 93,131)( 94,127)
( 95,129)( 96,128)( 97,134)( 98,133)( 99,135)(100,139)(101,141)(102,140)
(103,136)(104,138)(105,137)(106,143)(107,142)(108,144);;
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*s2*s1*s0*s1*s2*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*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(144)!( 73, 91)( 74, 92)( 75, 93)( 76, 94)( 77, 95)( 78, 96)( 79, 97)
( 80, 98)( 81, 99)( 82,100)( 83,101)( 84,102)( 85,103)( 86,104)( 87,105)
( 88,106)( 89,107)( 90,108)(109,127)(110,128)(111,129)(112,130)(113,131)
(114,132)(115,133)(116,134)(117,135)(118,136)(119,137)(120,138)(121,139)
(122,140)(123,141)(124,142)(125,143)(126,144);
s1 := Sym(144)!(  1, 73)(  2, 75)(  3, 74)(  4, 80)(  5, 79)(  6, 81)(  7, 77)
(  8, 76)(  9, 78)( 10, 82)( 11, 84)( 12, 83)( 13, 89)( 14, 88)( 15, 90)
( 16, 86)( 17, 85)( 18, 87)( 19, 91)( 20, 93)( 21, 92)( 22, 98)( 23, 97)
( 24, 99)( 25, 95)( 26, 94)( 27, 96)( 28,100)( 29,102)( 30,101)( 31,107)
( 32,106)( 33,108)( 34,104)( 35,103)( 36,105)( 37,118)( 38,120)( 39,119)
( 40,125)( 41,124)( 42,126)( 43,122)( 44,121)( 45,123)( 46,109)( 47,111)
( 48,110)( 49,116)( 50,115)( 51,117)( 52,113)( 53,112)( 54,114)( 55,136)
( 56,138)( 57,137)( 58,143)( 59,142)( 60,144)( 61,140)( 62,139)( 63,141)
( 64,127)( 65,129)( 66,128)( 67,134)( 68,133)( 69,135)( 70,131)( 71,130)
( 72,132);
s2 := Sym(144)!(  1,  4)(  2,  6)(  3,  5)(  7,  8)( 10, 13)( 11, 15)( 12, 14)
( 16, 17)( 19, 22)( 20, 24)( 21, 23)( 25, 26)( 28, 31)( 29, 33)( 30, 32)
( 34, 35)( 37, 49)( 38, 51)( 39, 50)( 40, 46)( 41, 48)( 42, 47)( 43, 53)
( 44, 52)( 45, 54)( 55, 67)( 56, 69)( 57, 68)( 58, 64)( 59, 66)( 60, 65)
( 61, 71)( 62, 70)( 63, 72)( 73,112)( 74,114)( 75,113)( 76,109)( 77,111)
( 78,110)( 79,116)( 80,115)( 81,117)( 82,121)( 83,123)( 84,122)( 85,118)
( 86,120)( 87,119)( 88,125)( 89,124)( 90,126)( 91,130)( 92,132)( 93,131)
( 94,127)( 95,129)( 96,128)( 97,134)( 98,133)( 99,135)(100,139)(101,141)
(102,140)(103,136)(104,138)(105,137)(106,143)(107,142)(108,144);
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, s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s2*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*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