# Polytope of Type {72,4}

Atlas Canonical Name : {72,4}*576b
if this polytope has a name.
Group : SmallGroup(576,309)
Rank : 3
Schlafli Type : {72,4}
Number of vertices, edges, etc : 72, 144, 4
Order of s0s1s2 : 72
Order of s0s1s2s1 : 4
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Flat
Self-Petrie
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Facet Of :
{72,4,2} of size 1152
Vertex Figure Of :
{2,72,4} of size 1152
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {36,4}*288a
3-fold quotients : {24,4}*192b
4-fold quotients : {36,2}*144, {18,4}*144a
6-fold quotients : {12,4}*96a
8-fold quotients : {18,2}*72
9-fold quotients : {8,4}*64b
12-fold quotients : {12,2}*48, {6,4}*48a
16-fold quotients : {9,2}*36
18-fold quotients : {4,4}*32
24-fold quotients : {6,2}*24
36-fold quotients : {2,4}*16, {4,2}*16
48-fold quotients : {3,2}*12
72-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {72,4}*1152a, {72,8}*1152a, {72,8}*1152d
3-fold covers : {216,4}*1728b, {72,12}*1728c, {72,12}*1728d
Permutation Representation (GAP) :
```s0 := (  2,  3)(  4,  8)(  5,  7)(  6,  9)( 11, 12)( 13, 17)( 14, 16)( 15, 18)
( 19, 28)( 20, 30)( 21, 29)( 22, 35)( 23, 34)( 24, 36)( 25, 32)( 26, 31)
( 27, 33)( 37, 46)( 38, 48)( 39, 47)( 40, 53)( 41, 52)( 42, 54)( 43, 50)
( 44, 49)( 45, 51)( 56, 57)( 58, 62)( 59, 61)( 60, 63)( 65, 66)( 67, 71)
( 68, 70)( 69, 72)( 73,109)( 74,111)( 75,110)( 76,116)( 77,115)( 78,117)
( 79,113)( 80,112)( 81,114)( 82,118)( 83,120)( 84,119)( 85,125)( 86,124)
( 87,126)( 88,122)( 89,121)( 90,123)( 91,136)( 92,138)( 93,137)( 94,143)
( 95,142)( 96,144)( 97,140)( 98,139)( 99,141)(100,127)(101,129)(102,128)
(103,134)(104,133)(105,135)(106,131)(107,130)(108,132);;
s1 := (  1, 76)(  2, 78)(  3, 77)(  4, 73)(  5, 75)(  6, 74)(  7, 80)(  8, 79)
(  9, 81)( 10, 85)( 11, 87)( 12, 86)( 13, 82)( 14, 84)( 15, 83)( 16, 89)
( 17, 88)( 18, 90)( 19, 94)( 20, 96)( 21, 95)( 22, 91)( 23, 93)( 24, 92)
( 25, 98)( 26, 97)( 27, 99)( 28,103)( 29,105)( 30,104)( 31,100)( 32,102)
( 33,101)( 34,107)( 35,106)( 36,108)( 37,121)( 38,123)( 39,122)( 40,118)
( 41,120)( 42,119)( 43,125)( 44,124)( 45,126)( 46,112)( 47,114)( 48,113)
( 49,109)( 50,111)( 51,110)( 52,116)( 53,115)( 54,117)( 55,139)( 56,141)
( 57,140)( 58,136)( 59,138)( 60,137)( 61,143)( 62,142)( 63,144)( 64,130)
( 65,132)( 66,131)( 67,127)( 68,129)( 69,128)( 70,134)( 71,133)( 72,135);;
s2 := ( 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);;
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*s2*s1*s2*s1*s0*s1*s0*s2*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*s2*s1*s0*s1*s0*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)!(  2,  3)(  4,  8)(  5,  7)(  6,  9)( 11, 12)( 13, 17)( 14, 16)
( 15, 18)( 19, 28)( 20, 30)( 21, 29)( 22, 35)( 23, 34)( 24, 36)( 25, 32)
( 26, 31)( 27, 33)( 37, 46)( 38, 48)( 39, 47)( 40, 53)( 41, 52)( 42, 54)
( 43, 50)( 44, 49)( 45, 51)( 56, 57)( 58, 62)( 59, 61)( 60, 63)( 65, 66)
( 67, 71)( 68, 70)( 69, 72)( 73,109)( 74,111)( 75,110)( 76,116)( 77,115)
( 78,117)( 79,113)( 80,112)( 81,114)( 82,118)( 83,120)( 84,119)( 85,125)
( 86,124)( 87,126)( 88,122)( 89,121)( 90,123)( 91,136)( 92,138)( 93,137)
( 94,143)( 95,142)( 96,144)( 97,140)( 98,139)( 99,141)(100,127)(101,129)
(102,128)(103,134)(104,133)(105,135)(106,131)(107,130)(108,132);
s1 := Sym(144)!(  1, 76)(  2, 78)(  3, 77)(  4, 73)(  5, 75)(  6, 74)(  7, 80)
(  8, 79)(  9, 81)( 10, 85)( 11, 87)( 12, 86)( 13, 82)( 14, 84)( 15, 83)
( 16, 89)( 17, 88)( 18, 90)( 19, 94)( 20, 96)( 21, 95)( 22, 91)( 23, 93)
( 24, 92)( 25, 98)( 26, 97)( 27, 99)( 28,103)( 29,105)( 30,104)( 31,100)
( 32,102)( 33,101)( 34,107)( 35,106)( 36,108)( 37,121)( 38,123)( 39,122)
( 40,118)( 41,120)( 42,119)( 43,125)( 44,124)( 45,126)( 46,112)( 47,114)
( 48,113)( 49,109)( 50,111)( 51,110)( 52,116)( 53,115)( 54,117)( 55,139)
( 56,141)( 57,140)( 58,136)( 59,138)( 60,137)( 61,143)( 62,142)( 63,144)
( 64,130)( 65,132)( 66,131)( 67,127)( 68,129)( 69,128)( 70,134)( 71,133)
( 72,135);
s2 := 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);
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*s2*s1*s2*s1*s0*s1*s0*s2*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*s2*s1*s0*s1*s0*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