# Polytope of Type {10,30}

Atlas Canonical Name : {10,30}*600b
Also Known As : {10,30|2}. if this polytope has another name.
Group : SmallGroup(600,195)
Rank : 3
Schlafli Type : {10,30}
Number of vertices, edges, etc : 10, 150, 30
Order of s0s1s2 : 30
Order of s0s1s2s1 : 2
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{10,30,2} of size 1200
Vertex Figure Of :
{2,10,30} of size 1200
Quotients (Maximal Quotients in Boldface) :
3-fold quotients : {10,10}*200a
5-fold quotients : {10,6}*120, {2,30}*120
10-fold quotients : {2,15}*60
15-fold quotients : {2,10}*40, {10,2}*40
25-fold quotients : {2,6}*24
30-fold quotients : {2,5}*20, {5,2}*20
50-fold quotients : {2,3}*12
75-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {20,30}*1200b, {10,60}*1200b
3-fold covers : {10,90}*1800b, {30,30}*1800c, {30,30}*1800g
Permutation Representation (GAP) :
```s0 := (  6, 21)(  7, 22)(  8, 23)(  9, 24)( 10, 25)( 11, 16)( 12, 17)( 13, 18)
( 14, 19)( 15, 20)( 31, 46)( 32, 47)( 33, 48)( 34, 49)( 35, 50)( 36, 41)
( 37, 42)( 38, 43)( 39, 44)( 40, 45)( 56, 71)( 57, 72)( 58, 73)( 59, 74)
( 60, 75)( 61, 66)( 62, 67)( 63, 68)( 64, 69)( 65, 70)( 81, 96)( 82, 97)
( 83, 98)( 84, 99)( 85,100)( 86, 91)( 87, 92)( 88, 93)( 89, 94)( 90, 95)
(106,121)(107,122)(108,123)(109,124)(110,125)(111,116)(112,117)(113,118)
(114,119)(115,120)(131,146)(132,147)(133,148)(134,149)(135,150)(136,141)
(137,142)(138,143)(139,144)(140,145);;
s1 := (  1,  6)(  2, 10)(  3,  9)(  4,  8)(  5,  7)( 11, 21)( 12, 25)( 13, 24)
( 14, 23)( 15, 22)( 17, 20)( 18, 19)( 26, 56)( 27, 60)( 28, 59)( 29, 58)
( 30, 57)( 31, 51)( 32, 55)( 33, 54)( 34, 53)( 35, 52)( 36, 71)( 37, 75)
( 38, 74)( 39, 73)( 40, 72)( 41, 66)( 42, 70)( 43, 69)( 44, 68)( 45, 67)
( 46, 61)( 47, 65)( 48, 64)( 49, 63)( 50, 62)( 76, 81)( 77, 85)( 78, 84)
( 79, 83)( 80, 82)( 86, 96)( 87,100)( 88, 99)( 89, 98)( 90, 97)( 92, 95)
( 93, 94)(101,131)(102,135)(103,134)(104,133)(105,132)(106,126)(107,130)
(108,129)(109,128)(110,127)(111,146)(112,150)(113,149)(114,148)(115,147)
(116,141)(117,145)(118,144)(119,143)(120,142)(121,136)(122,140)(123,139)
(124,138)(125,137);;
s2 := (  1,102)(  2,101)(  3,105)(  4,104)(  5,103)(  6,107)(  7,106)(  8,110)
(  9,109)( 10,108)( 11,112)( 12,111)( 13,115)( 14,114)( 15,113)( 16,117)
( 17,116)( 18,120)( 19,119)( 20,118)( 21,122)( 22,121)( 23,125)( 24,124)
( 25,123)( 26, 77)( 27, 76)( 28, 80)( 29, 79)( 30, 78)( 31, 82)( 32, 81)
( 33, 85)( 34, 84)( 35, 83)( 36, 87)( 37, 86)( 38, 90)( 39, 89)( 40, 88)
( 41, 92)( 42, 91)( 43, 95)( 44, 94)( 45, 93)( 46, 97)( 47, 96)( 48,100)
( 49, 99)( 50, 98)( 51,127)( 52,126)( 53,130)( 54,129)( 55,128)( 56,132)
( 57,131)( 58,135)( 59,134)( 60,133)( 61,137)( 62,136)( 63,140)( 64,139)
( 65,138)( 66,142)( 67,141)( 68,145)( 69,144)( 70,143)( 71,147)( 72,146)
( 73,150)( 74,149)( 75,148);;
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*s2*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,
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(150)!(  6, 21)(  7, 22)(  8, 23)(  9, 24)( 10, 25)( 11, 16)( 12, 17)
( 13, 18)( 14, 19)( 15, 20)( 31, 46)( 32, 47)( 33, 48)( 34, 49)( 35, 50)
( 36, 41)( 37, 42)( 38, 43)( 39, 44)( 40, 45)( 56, 71)( 57, 72)( 58, 73)
( 59, 74)( 60, 75)( 61, 66)( 62, 67)( 63, 68)( 64, 69)( 65, 70)( 81, 96)
( 82, 97)( 83, 98)( 84, 99)( 85,100)( 86, 91)( 87, 92)( 88, 93)( 89, 94)
( 90, 95)(106,121)(107,122)(108,123)(109,124)(110,125)(111,116)(112,117)
(113,118)(114,119)(115,120)(131,146)(132,147)(133,148)(134,149)(135,150)
(136,141)(137,142)(138,143)(139,144)(140,145);
s1 := Sym(150)!(  1,  6)(  2, 10)(  3,  9)(  4,  8)(  5,  7)( 11, 21)( 12, 25)
( 13, 24)( 14, 23)( 15, 22)( 17, 20)( 18, 19)( 26, 56)( 27, 60)( 28, 59)
( 29, 58)( 30, 57)( 31, 51)( 32, 55)( 33, 54)( 34, 53)( 35, 52)( 36, 71)
( 37, 75)( 38, 74)( 39, 73)( 40, 72)( 41, 66)( 42, 70)( 43, 69)( 44, 68)
( 45, 67)( 46, 61)( 47, 65)( 48, 64)( 49, 63)( 50, 62)( 76, 81)( 77, 85)
( 78, 84)( 79, 83)( 80, 82)( 86, 96)( 87,100)( 88, 99)( 89, 98)( 90, 97)
( 92, 95)( 93, 94)(101,131)(102,135)(103,134)(104,133)(105,132)(106,126)
(107,130)(108,129)(109,128)(110,127)(111,146)(112,150)(113,149)(114,148)
(115,147)(116,141)(117,145)(118,144)(119,143)(120,142)(121,136)(122,140)
(123,139)(124,138)(125,137);
s2 := Sym(150)!(  1,102)(  2,101)(  3,105)(  4,104)(  5,103)(  6,107)(  7,106)
(  8,110)(  9,109)( 10,108)( 11,112)( 12,111)( 13,115)( 14,114)( 15,113)
( 16,117)( 17,116)( 18,120)( 19,119)( 20,118)( 21,122)( 22,121)( 23,125)
( 24,124)( 25,123)( 26, 77)( 27, 76)( 28, 80)( 29, 79)( 30, 78)( 31, 82)
( 32, 81)( 33, 85)( 34, 84)( 35, 83)( 36, 87)( 37, 86)( 38, 90)( 39, 89)
( 40, 88)( 41, 92)( 42, 91)( 43, 95)( 44, 94)( 45, 93)( 46, 97)( 47, 96)
( 48,100)( 49, 99)( 50, 98)( 51,127)( 52,126)( 53,130)( 54,129)( 55,128)
( 56,132)( 57,131)( 58,135)( 59,134)( 60,133)( 61,137)( 62,136)( 63,140)
( 64,139)( 65,138)( 66,142)( 67,141)( 68,145)( 69,144)( 70,143)( 71,147)
( 72,146)( 73,150)( 74,149)( 75,148);
poly := sub<Sym(150)|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*s2*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,
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