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# Polytope of Type {4,30}

Atlas Canonical Name : {4,30}*240b
if this polytope has a name.
Group : SmallGroup(240,197)
Rank : 3
Schlafli Type : {4,30}
Number of vertices, edges, etc : 4, 60, 30
Order of s0s1s2 : 30
Order of s0s1s2s1 : 4
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Non-Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{4,30,2} of size 480
{4,30,4} of size 960
{4,30,4} of size 960
{4,30,4} of size 960
{4,30,6} of size 1440
{4,30,6} of size 1440
{4,30,8} of size 1920
{4,30,4} of size 1920
Vertex Figure Of :
{2,4,30} of size 480
{4,4,30} of size 1920
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {4,15}*120
5-fold quotients : {4,6}*48c
10-fold quotients : {4,3}*24
Covers (Minimal Covers in Boldface) :
2-fold covers : {4,60}*480b, {4,60}*480c, {4,30}*480
3-fold covers : {4,90}*720b
4-fold covers : {4,30}*960a, {4,120}*960c, {4,120}*960d, {4,60}*960b, {4,30}*960b, {4,60}*960c, {8,30}*960b, {8,30}*960c
5-fold covers : {4,150}*1200b
6-fold covers : {4,180}*1440b, {4,180}*1440c, {4,90}*1440, {12,30}*1440a, {12,30}*1440b
7-fold covers : {4,210}*1680b
8-fold covers : {4,60}*1920b, {4,60}*1920c, {8,30}*1920b, {8,30}*1920c, {4,240}*1920c, {4,240}*1920d, {4,60}*1920d, {8,60}*1920e, {8,60}*1920f, {4,30}*1920a, {8,30}*1920d, {8,30}*1920e, {8,30}*1920f, {8,60}*1920g, {8,60}*1920h, {4,120}*1920c, {4,120}*1920d, {8,30}*1920g, {4,60}*1920e, {4,120}*1920e, {4,30}*1920b, {4,120}*1920f
Permutation Representation (GAP) :
```s0 := (  1,  2)(  3,  4)(  5,  6)(  7,  8)(  9, 10)( 11, 12)( 13, 14)( 15, 16)
( 17, 18)( 19, 20)( 21, 22)( 23, 24)( 25, 26)( 27, 28)( 29, 30)( 31, 32)
( 33, 34)( 35, 36)( 37, 38)( 39, 40)( 41, 42)( 43, 44)( 45, 46)( 47, 48)
( 49, 50)( 51, 52)( 53, 54)( 55, 56)( 57, 58)( 59, 60)( 61, 62)( 63, 64)
( 65, 66)( 67, 68)( 69, 70)( 71, 72)( 73, 74)( 75, 76)( 77, 78)( 79, 80)
( 81, 82)( 83, 84)( 85, 86)( 87, 88)( 89, 90)( 91, 92)( 93, 94)( 95, 96)
( 97, 98)( 99,100)(101,102)(103,104)(105,106)(107,108)(109,110)(111,112)
(113,114)(115,116)(117,118)(119,120);;
s1 := (  2,  3)(  5, 17)(  6, 19)(  7, 18)(  8, 20)(  9, 13)( 10, 15)( 11, 14)
( 12, 16)( 21, 41)( 22, 43)( 23, 42)( 24, 44)( 25, 57)( 26, 59)( 27, 58)
( 28, 60)( 29, 53)( 30, 55)( 31, 54)( 32, 56)( 33, 49)( 34, 51)( 35, 50)
( 36, 52)( 37, 45)( 38, 47)( 39, 46)( 40, 48)( 62, 63)( 65, 77)( 66, 79)
( 67, 78)( 68, 80)( 69, 73)( 70, 75)( 71, 74)( 72, 76)( 81,101)( 82,103)
( 83,102)( 84,104)( 85,117)( 86,119)( 87,118)( 88,120)( 89,113)( 90,115)
( 91,114)( 92,116)( 93,109)( 94,111)( 95,110)( 96,112)( 97,105)( 98,107)
( 99,106)(100,108);;
s2 := (  1, 85)(  2, 86)(  3, 88)(  4, 87)(  5, 81)(  6, 82)(  7, 84)(  8, 83)
(  9, 97)( 10, 98)( 11,100)( 12, 99)( 13, 93)( 14, 94)( 15, 96)( 16, 95)
( 17, 89)( 18, 90)( 19, 92)( 20, 91)( 21, 65)( 22, 66)( 23, 68)( 24, 67)
( 25, 61)( 26, 62)( 27, 64)( 28, 63)( 29, 77)( 30, 78)( 31, 80)( 32, 79)
( 33, 73)( 34, 74)( 35, 76)( 36, 75)( 37, 69)( 38, 70)( 39, 72)( 40, 71)
( 41,105)( 42,106)( 43,108)( 44,107)( 45,101)( 46,102)( 47,104)( 48,103)
( 49,117)( 50,118)( 51,120)( 52,119)( 53,113)( 54,114)( 55,116)( 56,115)
( 57,109)( 58,110)( 59,112)( 60,111);;
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*s0*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 ];;
poly := F / rels;;

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

```
References : None.
to this polytope