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# Polytope of Type {4,72}

Atlas Canonical Name : {4,72}*576b
if this polytope has a name.
Group : SmallGroup(576,309)
Rank : 3
Schlafli Type : {4,72}
Number of vertices, edges, etc : 4, 144, 72
Order of s0s1s2 : 72
Order of s0s1s2s1 : 4
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
3-fold quotients : {4,24}*192b
4-fold quotients : {2,36}*144, {4,18}*144a
6-fold quotients : {4,12}*96a
8-fold quotients : {2,18}*72
9-fold quotients : {4,8}*64b
12-fold quotients : {2,12}*48, {4,6}*48a
16-fold quotients : {2,9}*36
18-fold quotients : {4,4}*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}*1152a, {8,72}*1152d
3-fold covers : {4,216}*1728b, {12,72}*1728c, {12,72}*1728d
Permutation Representation (GAP) :
```s0 := ( 37, 46)( 38, 47)( 39, 48)( 40, 49)( 41, 50)( 42, 51)( 43, 52)( 44, 53)
( 45, 54)( 55, 64)( 56, 65)( 57, 66)( 58, 67)( 59, 68)( 60, 69)( 61, 70)
( 62, 71)( 63, 72)( 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)(109,136)(110,137)(111,138)(112,139)
(113,140)(114,141)(115,142)(116,143)(117,144)(118,127)(119,128)(120,129)
(121,130)(122,131)(123,132)(124,133)(125,134)(126,135);;
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, 31)( 20, 33)( 21, 32)( 22, 28)( 23, 30)( 24, 29)( 25, 35)( 26, 34)
( 27, 36)( 37, 49)( 38, 51)( 39, 50)( 40, 46)( 41, 48)( 42, 47)( 43, 53)
( 44, 52)( 45, 54)( 55, 58)( 56, 60)( 57, 59)( 61, 62)( 64, 67)( 65, 69)
( 66, 68)( 70, 71)( 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)( 91,139)( 92,141)( 93,140)( 94,136)
( 95,138)( 96,137)( 97,143)( 98,142)( 99,144)(100,130)(101,132)(102,131)
(103,127)(104,129)(105,128)(106,134)(107,133)(108,135);;
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,
s2*s0*s1*s2*s1*s0*s1*s0*s2*s1*s2*s1*s0*s1,
s0*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*s0*s1*s2*s1 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := Sym(144)!( 37, 46)( 38, 47)( 39, 48)( 40, 49)( 41, 50)( 42, 51)( 43, 52)
( 44, 53)( 45, 54)( 55, 64)( 56, 65)( 57, 66)( 58, 67)( 59, 68)( 60, 69)
( 61, 70)( 62, 71)( 63, 72)( 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)(109,136)(110,137)(111,138)
(112,139)(113,140)(114,141)(115,142)(116,143)(117,144)(118,127)(119,128)
(120,129)(121,130)(122,131)(123,132)(124,133)(125,134)(126,135);
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, 31)( 20, 33)( 21, 32)( 22, 28)( 23, 30)( 24, 29)( 25, 35)
( 26, 34)( 27, 36)( 37, 49)( 38, 51)( 39, 50)( 40, 46)( 41, 48)( 42, 47)
( 43, 53)( 44, 52)( 45, 54)( 55, 58)( 56, 60)( 57, 59)( 61, 62)( 64, 67)
( 65, 69)( 66, 68)( 70, 71)( 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)( 91,139)( 92,141)( 93,140)
( 94,136)( 95,138)( 96,137)( 97,143)( 98,142)( 99,144)(100,130)(101,132)
(102,131)(103,127)(104,129)(105,128)(106,134)(107,133)(108,135);
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,
s2*s0*s1*s2*s1*s0*s1*s0*s2*s1*s2*s1*s0*s1,
s0*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*s0*s1*s2*s1 >;

```
References : None.
to this polytope