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# Polytope of Type {30,30}

Atlas Canonical Name : {30,30}*1800e
if this polytope has a name.
Group : SmallGroup(1800,678)
Rank : 3
Schlafli Type : {30,30}
Number of vertices, edges, etc : 30, 450, 30
Order of s0s1s2 : 30
Order of s0s1s2s1 : 30
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Facet Of :
None in this Atlas
Vertex Figure Of :
None in this Atlas
Quotients (Maximal Quotients in Boldface) :
3-fold quotients : {10,30}*600a
5-fold quotients : {30,6}*360a
9-fold quotients : {10,10}*200c
15-fold quotients : {10,6}*120
18-fold quotients : {5,10}*100
25-fold quotients : {6,6}*72b
45-fold quotients : {10,2}*40
50-fold quotients : {6,3}*36
75-fold quotients : {2,6}*24
90-fold quotients : {5,2}*20
150-fold quotients : {2,3}*12
225-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
```s0 := (  2,  5)(  3,  4)(  6, 21)(  7, 25)(  8, 24)(  9, 23)( 10, 22)( 11, 16)
( 12, 20)( 13, 19)( 14, 18)( 15, 17)( 27, 30)( 28, 29)( 31, 46)( 32, 50)
( 33, 49)( 34, 48)( 35, 47)( 36, 41)( 37, 45)( 38, 44)( 39, 43)( 40, 42)
( 52, 55)( 53, 54)( 56, 71)( 57, 75)( 58, 74)( 59, 73)( 60, 72)( 61, 66)
( 62, 70)( 63, 69)( 64, 68)( 65, 67)( 76,151)( 77,155)( 78,154)( 79,153)
( 80,152)( 81,171)( 82,175)( 83,174)( 84,173)( 85,172)( 86,166)( 87,170)
( 88,169)( 89,168)( 90,167)( 91,161)( 92,165)( 93,164)( 94,163)( 95,162)
( 96,156)( 97,160)( 98,159)( 99,158)(100,157)(101,176)(102,180)(103,179)
(104,178)(105,177)(106,196)(107,200)(108,199)(109,198)(110,197)(111,191)
(112,195)(113,194)(114,193)(115,192)(116,186)(117,190)(118,189)(119,188)
(120,187)(121,181)(122,185)(123,184)(124,183)(125,182)(126,201)(127,205)
(128,204)(129,203)(130,202)(131,221)(132,225)(133,224)(134,223)(135,222)
(136,216)(137,220)(138,219)(139,218)(140,217)(141,211)(142,215)(143,214)
(144,213)(145,212)(146,206)(147,210)(148,209)(149,208)(150,207);;
s1 := (  1, 82)(  2, 81)(  3, 85)(  4, 84)(  5, 83)(  6, 77)(  7, 76)(  8, 80)
(  9, 79)( 10, 78)( 11, 97)( 12, 96)( 13,100)( 14, 99)( 15, 98)( 16, 92)
( 17, 91)( 18, 95)( 19, 94)( 20, 93)( 21, 87)( 22, 86)( 23, 90)( 24, 89)
( 25, 88)( 26,132)( 27,131)( 28,135)( 29,134)( 30,133)( 31,127)( 32,126)
( 33,130)( 34,129)( 35,128)( 36,147)( 37,146)( 38,150)( 39,149)( 40,148)
( 41,142)( 42,141)( 43,145)( 44,144)( 45,143)( 46,137)( 47,136)( 48,140)
( 49,139)( 50,138)( 51,107)( 52,106)( 53,110)( 54,109)( 55,108)( 56,102)
( 57,101)( 58,105)( 59,104)( 60,103)( 61,122)( 62,121)( 63,125)( 64,124)
( 65,123)( 66,117)( 67,116)( 68,120)( 69,119)( 70,118)( 71,112)( 72,111)
( 73,115)( 74,114)( 75,113)(151,157)(152,156)(153,160)(154,159)(155,158)
(161,172)(162,171)(163,175)(164,174)(165,173)(166,167)(168,170)(176,207)
(177,206)(178,210)(179,209)(180,208)(181,202)(182,201)(183,205)(184,204)
(185,203)(186,222)(187,221)(188,225)(189,224)(190,223)(191,217)(192,216)
(193,220)(194,219)(195,218)(196,212)(197,211)(198,215)(199,214)(200,213);;
s2 := (  1, 26)(  2, 27)(  3, 28)(  4, 29)(  5, 30)(  6, 46)(  7, 47)(  8, 48)
(  9, 49)( 10, 50)( 11, 41)( 12, 42)( 13, 43)( 14, 44)( 15, 45)( 16, 36)
( 17, 37)( 18, 38)( 19, 39)( 20, 40)( 21, 31)( 22, 32)( 23, 33)( 24, 34)
( 25, 35)( 56, 71)( 57, 72)( 58, 73)( 59, 74)( 60, 75)( 61, 66)( 62, 67)
( 63, 68)( 64, 69)( 65, 70)( 76,176)( 77,177)( 78,178)( 79,179)( 80,180)
( 81,196)( 82,197)( 83,198)( 84,199)( 85,200)( 86,191)( 87,192)( 88,193)
( 89,194)( 90,195)( 91,186)( 92,187)( 93,188)( 94,189)( 95,190)( 96,181)
( 97,182)( 98,183)( 99,184)(100,185)(101,151)(102,152)(103,153)(104,154)
(105,155)(106,171)(107,172)(108,173)(109,174)(110,175)(111,166)(112,167)
(113,168)(114,169)(115,170)(116,161)(117,162)(118,163)(119,164)(120,165)
(121,156)(122,157)(123,158)(124,159)(125,160)(126,201)(127,202)(128,203)
(129,204)(130,205)(131,221)(132,222)(133,223)(134,224)(135,225)(136,216)
(137,217)(138,218)(139,219)(140,220)(141,211)(142,212)(143,213)(144,214)
(145,215)(146,206)(147,207)(148,208)(149,209)(150,210);;
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*s0*s1*s2*s0*s1*s2*s1*s0*s1,
s2*s1*s0*s2*s1*s0*s2*s1*s0*s1*s0*s1*s2*s0*s1*s0*s2*s1*s0*s2*s1*s0*s1*s0*s1*s0,
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*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(225)!(  2,  5)(  3,  4)(  6, 21)(  7, 25)(  8, 24)(  9, 23)( 10, 22)
( 11, 16)( 12, 20)( 13, 19)( 14, 18)( 15, 17)( 27, 30)( 28, 29)( 31, 46)
( 32, 50)( 33, 49)( 34, 48)( 35, 47)( 36, 41)( 37, 45)( 38, 44)( 39, 43)
( 40, 42)( 52, 55)( 53, 54)( 56, 71)( 57, 75)( 58, 74)( 59, 73)( 60, 72)
( 61, 66)( 62, 70)( 63, 69)( 64, 68)( 65, 67)( 76,151)( 77,155)( 78,154)
( 79,153)( 80,152)( 81,171)( 82,175)( 83,174)( 84,173)( 85,172)( 86,166)
( 87,170)( 88,169)( 89,168)( 90,167)( 91,161)( 92,165)( 93,164)( 94,163)
( 95,162)( 96,156)( 97,160)( 98,159)( 99,158)(100,157)(101,176)(102,180)
(103,179)(104,178)(105,177)(106,196)(107,200)(108,199)(109,198)(110,197)
(111,191)(112,195)(113,194)(114,193)(115,192)(116,186)(117,190)(118,189)
(119,188)(120,187)(121,181)(122,185)(123,184)(124,183)(125,182)(126,201)
(127,205)(128,204)(129,203)(130,202)(131,221)(132,225)(133,224)(134,223)
(135,222)(136,216)(137,220)(138,219)(139,218)(140,217)(141,211)(142,215)
(143,214)(144,213)(145,212)(146,206)(147,210)(148,209)(149,208)(150,207);
s1 := Sym(225)!(  1, 82)(  2, 81)(  3, 85)(  4, 84)(  5, 83)(  6, 77)(  7, 76)
(  8, 80)(  9, 79)( 10, 78)( 11, 97)( 12, 96)( 13,100)( 14, 99)( 15, 98)
( 16, 92)( 17, 91)( 18, 95)( 19, 94)( 20, 93)( 21, 87)( 22, 86)( 23, 90)
( 24, 89)( 25, 88)( 26,132)( 27,131)( 28,135)( 29,134)( 30,133)( 31,127)
( 32,126)( 33,130)( 34,129)( 35,128)( 36,147)( 37,146)( 38,150)( 39,149)
( 40,148)( 41,142)( 42,141)( 43,145)( 44,144)( 45,143)( 46,137)( 47,136)
( 48,140)( 49,139)( 50,138)( 51,107)( 52,106)( 53,110)( 54,109)( 55,108)
( 56,102)( 57,101)( 58,105)( 59,104)( 60,103)( 61,122)( 62,121)( 63,125)
( 64,124)( 65,123)( 66,117)( 67,116)( 68,120)( 69,119)( 70,118)( 71,112)
( 72,111)( 73,115)( 74,114)( 75,113)(151,157)(152,156)(153,160)(154,159)
(155,158)(161,172)(162,171)(163,175)(164,174)(165,173)(166,167)(168,170)
(176,207)(177,206)(178,210)(179,209)(180,208)(181,202)(182,201)(183,205)
(184,204)(185,203)(186,222)(187,221)(188,225)(189,224)(190,223)(191,217)
(192,216)(193,220)(194,219)(195,218)(196,212)(197,211)(198,215)(199,214)
(200,213);
s2 := Sym(225)!(  1, 26)(  2, 27)(  3, 28)(  4, 29)(  5, 30)(  6, 46)(  7, 47)
(  8, 48)(  9, 49)( 10, 50)( 11, 41)( 12, 42)( 13, 43)( 14, 44)( 15, 45)
( 16, 36)( 17, 37)( 18, 38)( 19, 39)( 20, 40)( 21, 31)( 22, 32)( 23, 33)
( 24, 34)( 25, 35)( 56, 71)( 57, 72)( 58, 73)( 59, 74)( 60, 75)( 61, 66)
( 62, 67)( 63, 68)( 64, 69)( 65, 70)( 76,176)( 77,177)( 78,178)( 79,179)
( 80,180)( 81,196)( 82,197)( 83,198)( 84,199)( 85,200)( 86,191)( 87,192)
( 88,193)( 89,194)( 90,195)( 91,186)( 92,187)( 93,188)( 94,189)( 95,190)
( 96,181)( 97,182)( 98,183)( 99,184)(100,185)(101,151)(102,152)(103,153)
(104,154)(105,155)(106,171)(107,172)(108,173)(109,174)(110,175)(111,166)
(112,167)(113,168)(114,169)(115,170)(116,161)(117,162)(118,163)(119,164)
(120,165)(121,156)(122,157)(123,158)(124,159)(125,160)(126,201)(127,202)
(128,203)(129,204)(130,205)(131,221)(132,222)(133,223)(134,224)(135,225)
(136,216)(137,217)(138,218)(139,219)(140,220)(141,211)(142,212)(143,213)
(144,214)(145,215)(146,206)(147,207)(148,208)(149,209)(150,210);
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*s0*s1*s2*s0*s1*s2*s1*s0*s1,
s2*s1*s0*s2*s1*s0*s2*s1*s0*s1*s0*s1*s2*s0*s1*s0*s2*s1*s0*s2*s1*s0*s1*s0*s1*s0,
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*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