# Polytope of Type {18,10}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {18,10}*900
if this polytope has a name.
Group : SmallGroup(900,48)
Rank : 3
Schlafli Type : {18,10}
Number of vertices, edges, etc : 45, 225, 25
Order of s0s1s2 : 9
Order of s0s1s2s1 : 10
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Non-Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Facet Of :
{18,10,2} of size 1800
Vertex Figure Of :
{2,18,10} of size 1800
Quotients (Maximal Quotients in Boldface) :
3-fold quotients : {6,10}*300
Covers (Minimal Covers in Boldface) :
2-fold covers : {18,10}*1800a
Permutation Representation (GAP) :
```s0 := (  2, 11)(  3, 21)(  4,  6)(  5, 16)(  7, 14)(  8, 24)( 10, 19)( 13, 22)
( 15, 17)( 18, 25)( 26, 51)( 27, 61)( 28, 71)( 29, 56)( 30, 66)( 31, 54)
( 32, 64)( 33, 74)( 34, 59)( 35, 69)( 36, 52)( 37, 62)( 38, 72)( 39, 57)
( 40, 67)( 41, 55)( 42, 65)( 43, 75)( 44, 60)( 45, 70)( 46, 53)( 47, 63)
( 48, 73)( 49, 58)( 50, 68)( 76,176)( 77,186)( 78,196)( 79,181)( 80,191)
( 81,179)( 82,189)( 83,199)( 84,184)( 85,194)( 86,177)( 87,187)( 88,197)
( 89,182)( 90,192)( 91,180)( 92,190)( 93,200)( 94,185)( 95,195)( 96,178)
( 97,188)( 98,198)( 99,183)(100,193)(101,151)(102,161)(103,171)(104,156)
(105,166)(106,154)(107,164)(108,174)(109,159)(110,169)(111,152)(112,162)
(113,172)(114,157)(115,167)(116,155)(117,165)(118,175)(119,160)(120,170)
(121,153)(122,163)(123,173)(124,158)(125,168)(126,201)(127,211)(128,221)
(129,206)(130,216)(131,204)(132,214)(133,224)(134,209)(135,219)(136,202)
(137,212)(138,222)(139,207)(140,217)(141,205)(142,215)(143,225)(144,210)
(145,220)(146,203)(147,213)(148,223)(149,208)(150,218);;
s1 := (  1, 76)(  2, 87)(  3, 98)(  4, 84)(  5, 95)(  6, 96)(  7, 82)(  8, 93)
(  9, 79)( 10, 90)( 11, 91)( 12, 77)( 13, 88)( 14, 99)( 15, 85)( 16, 86)
( 17, 97)( 18, 83)( 19, 94)( 20, 80)( 21, 81)( 22, 92)( 23, 78)( 24, 89)
( 25,100)( 26,126)( 27,137)( 28,148)( 29,134)( 30,145)( 31,146)( 32,132)
( 33,143)( 34,129)( 35,140)( 36,141)( 37,127)( 38,138)( 39,149)( 40,135)
( 41,136)( 42,147)( 43,133)( 44,144)( 45,130)( 46,131)( 47,142)( 48,128)
( 49,139)( 50,150)( 51,101)( 52,112)( 53,123)( 54,109)( 55,120)( 56,121)
( 57,107)( 58,118)( 59,104)( 60,115)( 61,116)( 62,102)( 63,113)( 64,124)
( 65,110)( 66,111)( 67,122)( 68,108)( 69,119)( 70,105)( 71,106)( 72,117)
( 73,103)( 74,114)( 75,125)(151,176)(152,187)(153,198)(154,184)(155,195)
(156,196)(157,182)(158,193)(159,179)(160,190)(161,191)(162,177)(163,188)
(164,199)(165,185)(166,186)(167,197)(168,183)(169,194)(170,180)(171,181)
(172,192)(173,178)(174,189)(175,200)(202,212)(203,223)(204,209)(205,220)
(206,221)(208,218)(210,215)(211,216)(214,224)(217,222);;
s2 := (  1, 12)(  2, 11)(  3, 15)(  4, 14)(  5, 13)(  6,  7)(  8, 10)( 16, 22)
( 17, 21)( 18, 25)( 19, 24)( 20, 23)( 26, 37)( 27, 36)( 28, 40)( 29, 39)
( 30, 38)( 31, 32)( 33, 35)( 41, 47)( 42, 46)( 43, 50)( 44, 49)( 45, 48)
( 51, 62)( 52, 61)( 53, 65)( 54, 64)( 55, 63)( 56, 57)( 58, 60)( 66, 72)
( 67, 71)( 68, 75)( 69, 74)( 70, 73)( 76, 87)( 77, 86)( 78, 90)( 79, 89)
( 80, 88)( 81, 82)( 83, 85)( 91, 97)( 92, 96)( 93,100)( 94, 99)( 95, 98)
(101,112)(102,111)(103,115)(104,114)(105,113)(106,107)(108,110)(116,122)
(117,121)(118,125)(119,124)(120,123)(126,137)(127,136)(128,140)(129,139)
(130,138)(131,132)(133,135)(141,147)(142,146)(143,150)(144,149)(145,148)
(151,162)(152,161)(153,165)(154,164)(155,163)(156,157)(158,160)(166,172)
(167,171)(168,175)(169,174)(170,173)(176,187)(177,186)(178,190)(179,189)
(180,188)(181,182)(183,185)(191,197)(192,196)(193,200)(194,199)(195,198)
(201,212)(202,211)(203,215)(204,214)(205,213)(206,207)(208,210)(216,222)
(217,221)(218,225)(219,224)(220,223);;
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, s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1,
s0*s1*s2*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*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,
s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := Sym(225)!(  2, 11)(  3, 21)(  4,  6)(  5, 16)(  7, 14)(  8, 24)( 10, 19)
( 13, 22)( 15, 17)( 18, 25)( 26, 51)( 27, 61)( 28, 71)( 29, 56)( 30, 66)
( 31, 54)( 32, 64)( 33, 74)( 34, 59)( 35, 69)( 36, 52)( 37, 62)( 38, 72)
( 39, 57)( 40, 67)( 41, 55)( 42, 65)( 43, 75)( 44, 60)( 45, 70)( 46, 53)
( 47, 63)( 48, 73)( 49, 58)( 50, 68)( 76,176)( 77,186)( 78,196)( 79,181)
( 80,191)( 81,179)( 82,189)( 83,199)( 84,184)( 85,194)( 86,177)( 87,187)
( 88,197)( 89,182)( 90,192)( 91,180)( 92,190)( 93,200)( 94,185)( 95,195)
( 96,178)( 97,188)( 98,198)( 99,183)(100,193)(101,151)(102,161)(103,171)
(104,156)(105,166)(106,154)(107,164)(108,174)(109,159)(110,169)(111,152)
(112,162)(113,172)(114,157)(115,167)(116,155)(117,165)(118,175)(119,160)
(120,170)(121,153)(122,163)(123,173)(124,158)(125,168)(126,201)(127,211)
(128,221)(129,206)(130,216)(131,204)(132,214)(133,224)(134,209)(135,219)
(136,202)(137,212)(138,222)(139,207)(140,217)(141,205)(142,215)(143,225)
(144,210)(145,220)(146,203)(147,213)(148,223)(149,208)(150,218);
s1 := Sym(225)!(  1, 76)(  2, 87)(  3, 98)(  4, 84)(  5, 95)(  6, 96)(  7, 82)
(  8, 93)(  9, 79)( 10, 90)( 11, 91)( 12, 77)( 13, 88)( 14, 99)( 15, 85)
( 16, 86)( 17, 97)( 18, 83)( 19, 94)( 20, 80)( 21, 81)( 22, 92)( 23, 78)
( 24, 89)( 25,100)( 26,126)( 27,137)( 28,148)( 29,134)( 30,145)( 31,146)
( 32,132)( 33,143)( 34,129)( 35,140)( 36,141)( 37,127)( 38,138)( 39,149)
( 40,135)( 41,136)( 42,147)( 43,133)( 44,144)( 45,130)( 46,131)( 47,142)
( 48,128)( 49,139)( 50,150)( 51,101)( 52,112)( 53,123)( 54,109)( 55,120)
( 56,121)( 57,107)( 58,118)( 59,104)( 60,115)( 61,116)( 62,102)( 63,113)
( 64,124)( 65,110)( 66,111)( 67,122)( 68,108)( 69,119)( 70,105)( 71,106)
( 72,117)( 73,103)( 74,114)( 75,125)(151,176)(152,187)(153,198)(154,184)
(155,195)(156,196)(157,182)(158,193)(159,179)(160,190)(161,191)(162,177)
(163,188)(164,199)(165,185)(166,186)(167,197)(168,183)(169,194)(170,180)
(171,181)(172,192)(173,178)(174,189)(175,200)(202,212)(203,223)(204,209)
(205,220)(206,221)(208,218)(210,215)(211,216)(214,224)(217,222);
s2 := Sym(225)!(  1, 12)(  2, 11)(  3, 15)(  4, 14)(  5, 13)(  6,  7)(  8, 10)
( 16, 22)( 17, 21)( 18, 25)( 19, 24)( 20, 23)( 26, 37)( 27, 36)( 28, 40)
( 29, 39)( 30, 38)( 31, 32)( 33, 35)( 41, 47)( 42, 46)( 43, 50)( 44, 49)
( 45, 48)( 51, 62)( 52, 61)( 53, 65)( 54, 64)( 55, 63)( 56, 57)( 58, 60)
( 66, 72)( 67, 71)( 68, 75)( 69, 74)( 70, 73)( 76, 87)( 77, 86)( 78, 90)
( 79, 89)( 80, 88)( 81, 82)( 83, 85)( 91, 97)( 92, 96)( 93,100)( 94, 99)
( 95, 98)(101,112)(102,111)(103,115)(104,114)(105,113)(106,107)(108,110)
(116,122)(117,121)(118,125)(119,124)(120,123)(126,137)(127,136)(128,140)
(129,139)(130,138)(131,132)(133,135)(141,147)(142,146)(143,150)(144,149)
(145,148)(151,162)(152,161)(153,165)(154,164)(155,163)(156,157)(158,160)
(166,172)(167,171)(168,175)(169,174)(170,173)(176,187)(177,186)(178,190)
(179,189)(180,188)(181,182)(183,185)(191,197)(192,196)(193,200)(194,199)
(195,198)(201,212)(202,211)(203,215)(204,214)(205,213)(206,207)(208,210)
(216,222)(217,221)(218,225)(219,224)(220,223);
poly := sub<Sym(225)|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, s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1,
s0*s1*s2*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*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,
s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;

```
References : None.
to this polytope