Polytope of Type {30,30}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {30,30}*1800c
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 : 6
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}*600b
   5-fold quotients : {30,6}*360a, {6,30}*360c
   9-fold quotients : {10,10}*200a
   10-fold quotients : {6,15}*180
   15-fold quotients : {10,6}*120, {2,30}*120
   25-fold quotients : {6,6}*72b
   30-fold quotients : {2,15}*60
   45-fold quotients : {2,10}*40, {10,2}*40
   50-fold quotients : {6,3}*36
   75-fold quotients : {2,6}*24
   90-fold quotients : {2,5}*20, {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)(  7, 10)(  8,  9)( 12, 15)( 13, 14)( 17, 20)( 18, 19)
( 22, 25)( 23, 24)( 26, 51)( 27, 55)( 28, 54)( 29, 53)( 30, 52)( 31, 56)
( 32, 60)( 33, 59)( 34, 58)( 35, 57)( 36, 61)( 37, 65)( 38, 64)( 39, 63)
( 40, 62)( 41, 66)( 42, 70)( 43, 69)( 44, 68)( 45, 67)( 46, 71)( 47, 75)
( 48, 74)( 49, 73)( 50, 72)( 77, 80)( 78, 79)( 82, 85)( 83, 84)( 87, 90)
( 88, 89)( 92, 95)( 93, 94)( 97,100)( 98, 99)(101,126)(102,130)(103,129)
(104,128)(105,127)(106,131)(107,135)(108,134)(109,133)(110,132)(111,136)
(112,140)(113,139)(114,138)(115,137)(116,141)(117,145)(118,144)(119,143)
(120,142)(121,146)(122,150)(123,149)(124,148)(125,147)(152,155)(153,154)
(157,160)(158,159)(162,165)(163,164)(167,170)(168,169)(172,175)(173,174)
(176,201)(177,205)(178,204)(179,203)(180,202)(181,206)(182,210)(183,209)
(184,208)(185,207)(186,211)(187,215)(188,214)(189,213)(190,212)(191,216)
(192,220)(193,219)(194,218)(195,217)(196,221)(197,225)(198,224)(199,223)
(200,222);;
s1 := (  1, 27)(  2, 26)(  3, 30)(  4, 29)(  5, 28)(  6, 47)(  7, 46)(  8, 50)
(  9, 49)( 10, 48)( 11, 42)( 12, 41)( 13, 45)( 14, 44)( 15, 43)( 16, 37)
( 17, 36)( 18, 40)( 19, 39)( 20, 38)( 21, 32)( 22, 31)( 23, 35)( 24, 34)
( 25, 33)( 51, 52)( 53, 55)( 56, 72)( 57, 71)( 58, 75)( 59, 74)( 60, 73)
( 61, 67)( 62, 66)( 63, 70)( 64, 69)( 65, 68)( 76,177)( 77,176)( 78,180)
( 79,179)( 80,178)( 81,197)( 82,196)( 83,200)( 84,199)( 85,198)( 86,192)
( 87,191)( 88,195)( 89,194)( 90,193)( 91,187)( 92,186)( 93,190)( 94,189)
( 95,188)( 96,182)( 97,181)( 98,185)( 99,184)(100,183)(101,152)(102,151)
(103,155)(104,154)(105,153)(106,172)(107,171)(108,175)(109,174)(110,173)
(111,167)(112,166)(113,170)(114,169)(115,168)(116,162)(117,161)(118,165)
(119,164)(120,163)(121,157)(122,156)(123,160)(124,159)(125,158)(126,202)
(127,201)(128,205)(129,204)(130,203)(131,222)(132,221)(133,225)(134,224)
(135,223)(136,217)(137,216)(138,220)(139,219)(140,218)(141,212)(142,211)
(143,215)(144,214)(145,213)(146,207)(147,206)(148,210)(149,209)(150,208);;
s2 := (  1, 81)(  2, 82)(  3, 83)(  4, 84)(  5, 85)(  6, 76)(  7, 77)(  8, 78)
(  9, 79)( 10, 80)( 11, 96)( 12, 97)( 13, 98)( 14, 99)( 15,100)( 16, 91)
( 17, 92)( 18, 93)( 19, 94)( 20, 95)( 21, 86)( 22, 87)( 23, 88)( 24, 89)
( 25, 90)( 26,131)( 27,132)( 28,133)( 29,134)( 30,135)( 31,126)( 32,127)
( 33,128)( 34,129)( 35,130)( 36,146)( 37,147)( 38,148)( 39,149)( 40,150)
( 41,141)( 42,142)( 43,143)( 44,144)( 45,145)( 46,136)( 47,137)( 48,138)
( 49,139)( 50,140)( 51,106)( 52,107)( 53,108)( 54,109)( 55,110)( 56,101)
( 57,102)( 58,103)( 59,104)( 60,105)( 61,121)( 62,122)( 63,123)( 64,124)
( 65,125)( 66,116)( 67,117)( 68,118)( 69,119)( 70,120)( 71,111)( 72,112)
( 73,113)( 74,114)( 75,115)(151,156)(152,157)(153,158)(154,159)(155,160)
(161,171)(162,172)(163,173)(164,174)(165,175)(176,206)(177,207)(178,208)
(179,209)(180,210)(181,201)(182,202)(183,203)(184,204)(185,205)(186,221)
(187,222)(188,223)(189,224)(190,225)(191,216)(192,217)(193,218)(194,219)
(195,220)(196,211)(197,212)(198,213)(199,214)(200,215);;
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, 
s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*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(225)!(  2,  5)(  3,  4)(  7, 10)(  8,  9)( 12, 15)( 13, 14)( 17, 20)
( 18, 19)( 22, 25)( 23, 24)( 26, 51)( 27, 55)( 28, 54)( 29, 53)( 30, 52)
( 31, 56)( 32, 60)( 33, 59)( 34, 58)( 35, 57)( 36, 61)( 37, 65)( 38, 64)
( 39, 63)( 40, 62)( 41, 66)( 42, 70)( 43, 69)( 44, 68)( 45, 67)( 46, 71)
( 47, 75)( 48, 74)( 49, 73)( 50, 72)( 77, 80)( 78, 79)( 82, 85)( 83, 84)
( 87, 90)( 88, 89)( 92, 95)( 93, 94)( 97,100)( 98, 99)(101,126)(102,130)
(103,129)(104,128)(105,127)(106,131)(107,135)(108,134)(109,133)(110,132)
(111,136)(112,140)(113,139)(114,138)(115,137)(116,141)(117,145)(118,144)
(119,143)(120,142)(121,146)(122,150)(123,149)(124,148)(125,147)(152,155)
(153,154)(157,160)(158,159)(162,165)(163,164)(167,170)(168,169)(172,175)
(173,174)(176,201)(177,205)(178,204)(179,203)(180,202)(181,206)(182,210)
(183,209)(184,208)(185,207)(186,211)(187,215)(188,214)(189,213)(190,212)
(191,216)(192,220)(193,219)(194,218)(195,217)(196,221)(197,225)(198,224)
(199,223)(200,222);
s1 := Sym(225)!(  1, 27)(  2, 26)(  3, 30)(  4, 29)(  5, 28)(  6, 47)(  7, 46)
(  8, 50)(  9, 49)( 10, 48)( 11, 42)( 12, 41)( 13, 45)( 14, 44)( 15, 43)
( 16, 37)( 17, 36)( 18, 40)( 19, 39)( 20, 38)( 21, 32)( 22, 31)( 23, 35)
( 24, 34)( 25, 33)( 51, 52)( 53, 55)( 56, 72)( 57, 71)( 58, 75)( 59, 74)
( 60, 73)( 61, 67)( 62, 66)( 63, 70)( 64, 69)( 65, 68)( 76,177)( 77,176)
( 78,180)( 79,179)( 80,178)( 81,197)( 82,196)( 83,200)( 84,199)( 85,198)
( 86,192)( 87,191)( 88,195)( 89,194)( 90,193)( 91,187)( 92,186)( 93,190)
( 94,189)( 95,188)( 96,182)( 97,181)( 98,185)( 99,184)(100,183)(101,152)
(102,151)(103,155)(104,154)(105,153)(106,172)(107,171)(108,175)(109,174)
(110,173)(111,167)(112,166)(113,170)(114,169)(115,168)(116,162)(117,161)
(118,165)(119,164)(120,163)(121,157)(122,156)(123,160)(124,159)(125,158)
(126,202)(127,201)(128,205)(129,204)(130,203)(131,222)(132,221)(133,225)
(134,224)(135,223)(136,217)(137,216)(138,220)(139,219)(140,218)(141,212)
(142,211)(143,215)(144,214)(145,213)(146,207)(147,206)(148,210)(149,209)
(150,208);
s2 := Sym(225)!(  1, 81)(  2, 82)(  3, 83)(  4, 84)(  5, 85)(  6, 76)(  7, 77)
(  8, 78)(  9, 79)( 10, 80)( 11, 96)( 12, 97)( 13, 98)( 14, 99)( 15,100)
( 16, 91)( 17, 92)( 18, 93)( 19, 94)( 20, 95)( 21, 86)( 22, 87)( 23, 88)
( 24, 89)( 25, 90)( 26,131)( 27,132)( 28,133)( 29,134)( 30,135)( 31,126)
( 32,127)( 33,128)( 34,129)( 35,130)( 36,146)( 37,147)( 38,148)( 39,149)
( 40,150)( 41,141)( 42,142)( 43,143)( 44,144)( 45,145)( 46,136)( 47,137)
( 48,138)( 49,139)( 50,140)( 51,106)( 52,107)( 53,108)( 54,109)( 55,110)
( 56,101)( 57,102)( 58,103)( 59,104)( 60,105)( 61,121)( 62,122)( 63,123)
( 64,124)( 65,125)( 66,116)( 67,117)( 68,118)( 69,119)( 70,120)( 71,111)
( 72,112)( 73,113)( 74,114)( 75,115)(151,156)(152,157)(153,158)(154,159)
(155,160)(161,171)(162,172)(163,173)(164,174)(165,175)(176,206)(177,207)
(178,208)(179,209)(180,210)(181,201)(182,202)(183,203)(184,204)(185,205)
(186,221)(187,222)(188,223)(189,224)(190,225)(191,216)(192,217)(193,218)
(194,219)(195,220)(196,211)(197,212)(198,213)(199,214)(200,215);
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, 
s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*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