Polytope of Type {30,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {30,4}*1800
if this polytope has a name.
Group : SmallGroup(1800,664)
Rank : 3
Schlafli Type : {30,4}
Number of vertices, edges, etc : 225, 450, 30
Order of s₀s₁s₂ : 4
Order of s₀s₁s₂s₁ : 30
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Non-Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
   Skewing Operation
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   9-fold quotients : {10,4}*200
   25-fold quotients : {6,4}*72
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)( 26, 51)( 27, 55)( 28, 54)( 29, 53)
( 30, 52)( 31, 71)( 32, 75)( 33, 74)( 34, 73)( 35, 72)( 36, 66)( 37, 70)
( 38, 69)( 39, 68)( 40, 67)( 41, 61)( 42, 65)( 43, 64)( 44, 63)( 45, 62)
( 46, 56)( 47, 60)( 48, 59)( 49, 58)( 50, 57)( 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)(101,201)(102,205)
(103,204)(104,203)(105,202)(106,221)(107,225)(108,224)(109,223)(110,222)
(111,216)(112,220)(113,219)(114,218)(115,217)(116,211)(117,215)(118,214)
(119,213)(120,212)(121,206)(122,210)(123,209)(124,208)(125,207)(126,176)
(127,180)(128,179)(129,178)(130,177)(131,196)(132,200)(133,199)(134,198)
(135,197)(136,191)(137,195)(138,194)(139,193)(140,192)(141,186)(142,190)
(143,189)(144,188)(145,187)(146,181)(147,185)(148,184)(149,183)(150,182);;
s1 := (  1, 81)(  2, 92)(  3, 78)(  4, 89)(  5,100)(  6, 76)(  7, 87)(  8, 98)
(  9, 84)( 10, 95)( 11, 96)( 12, 82)( 13, 93)( 14, 79)( 15, 90)( 16, 91)
( 17, 77)( 18, 88)( 19, 99)( 20, 85)( 21, 86)( 22, 97)( 23, 83)( 24, 94)
( 25, 80)( 26,106)( 27,117)( 28,103)( 29,114)( 30,125)( 31,101)( 32,112)
( 33,123)( 34,109)( 35,120)( 36,121)( 37,107)( 38,118)( 39,104)( 40,115)
( 41,116)( 42,102)( 43,113)( 44,124)( 45,110)( 46,111)( 47,122)( 48,108)
( 49,119)( 50,105)( 51,131)( 52,142)( 53,128)( 54,139)( 55,150)( 56,126)
( 57,137)( 58,148)( 59,134)( 60,145)( 61,146)( 62,132)( 63,143)( 64,129)
( 65,140)( 66,141)( 67,127)( 68,138)( 69,149)( 70,135)( 71,136)( 72,147)
( 73,133)( 74,144)( 75,130)(151,156)(152,167)(154,164)(155,175)(157,162)
(158,173)(160,170)(161,171)(163,168)(169,174)(176,181)(177,192)(179,189)
(180,200)(182,187)(183,198)(185,195)(186,196)(188,193)(194,199)(201,206)
(202,217)(204,214)(205,225)(207,212)(208,223)(210,220)(211,221)(213,218)
(219,224);;
s2 := (  2,  9)(  3, 12)(  4, 20)(  5, 23)(  6, 13)(  7, 16)(  8, 24)( 11, 25)
( 15, 17)( 19, 21)( 26,151)( 27,159)( 28,162)( 29,170)( 30,173)( 31,163)
( 32,166)( 33,174)( 34,152)( 35,160)( 36,175)( 37,153)( 38,156)( 39,164)
( 40,167)( 41,157)( 42,165)( 43,168)( 44,171)( 45,154)( 46,169)( 47,172)
( 48,155)( 49,158)( 50,161)( 51, 76)( 52, 84)( 53, 87)( 54, 95)( 55, 98)
( 56, 88)( 57, 91)( 58, 99)( 59, 77)( 60, 85)( 61,100)( 62, 78)( 63, 81)
( 64, 89)( 65, 92)( 66, 82)( 67, 90)( 68, 93)( 69, 96)( 70, 79)( 71, 94)
( 72, 97)( 73, 80)( 74, 83)( 75, 86)(101,201)(102,209)(103,212)(104,220)
(105,223)(106,213)(107,216)(108,224)(109,202)(110,210)(111,225)(112,203)
(113,206)(114,214)(115,217)(116,207)(117,215)(118,218)(119,221)(120,204)
(121,219)(122,222)(123,205)(124,208)(125,211)(127,134)(128,137)(129,145)
(130,148)(131,138)(132,141)(133,149)(136,150)(140,142)(144,146)(177,184)
(178,187)(179,195)(180,198)(181,188)(182,191)(183,199)(186,200)(190,192)
(194,196);;
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, s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*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*s0*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*s0*s1*s0*s1*s0*s1*s0*s1*s2*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)( 26, 51)( 27, 55)( 28, 54)
( 29, 53)( 30, 52)( 31, 71)( 32, 75)( 33, 74)( 34, 73)( 35, 72)( 36, 66)
( 37, 70)( 38, 69)( 39, 68)( 40, 67)( 41, 61)( 42, 65)( 43, 64)( 44, 63)
( 45, 62)( 46, 56)( 47, 60)( 48, 59)( 49, 58)( 50, 57)( 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)(101,201)
(102,205)(103,204)(104,203)(105,202)(106,221)(107,225)(108,224)(109,223)
(110,222)(111,216)(112,220)(113,219)(114,218)(115,217)(116,211)(117,215)
(118,214)(119,213)(120,212)(121,206)(122,210)(123,209)(124,208)(125,207)
(126,176)(127,180)(128,179)(129,178)(130,177)(131,196)(132,200)(133,199)
(134,198)(135,197)(136,191)(137,195)(138,194)(139,193)(140,192)(141,186)
(142,190)(143,189)(144,188)(145,187)(146,181)(147,185)(148,184)(149,183)
(150,182);
s1 := Sym(225)!(  1, 81)(  2, 92)(  3, 78)(  4, 89)(  5,100)(  6, 76)(  7, 87)
(  8, 98)(  9, 84)( 10, 95)( 11, 96)( 12, 82)( 13, 93)( 14, 79)( 15, 90)
( 16, 91)( 17, 77)( 18, 88)( 19, 99)( 20, 85)( 21, 86)( 22, 97)( 23, 83)
( 24, 94)( 25, 80)( 26,106)( 27,117)( 28,103)( 29,114)( 30,125)( 31,101)
( 32,112)( 33,123)( 34,109)( 35,120)( 36,121)( 37,107)( 38,118)( 39,104)
( 40,115)( 41,116)( 42,102)( 43,113)( 44,124)( 45,110)( 46,111)( 47,122)
( 48,108)( 49,119)( 50,105)( 51,131)( 52,142)( 53,128)( 54,139)( 55,150)
( 56,126)( 57,137)( 58,148)( 59,134)( 60,145)( 61,146)( 62,132)( 63,143)
( 64,129)( 65,140)( 66,141)( 67,127)( 68,138)( 69,149)( 70,135)( 71,136)
( 72,147)( 73,133)( 74,144)( 75,130)(151,156)(152,167)(154,164)(155,175)
(157,162)(158,173)(160,170)(161,171)(163,168)(169,174)(176,181)(177,192)
(179,189)(180,200)(182,187)(183,198)(185,195)(186,196)(188,193)(194,199)
(201,206)(202,217)(204,214)(205,225)(207,212)(208,223)(210,220)(211,221)
(213,218)(219,224);
s2 := Sym(225)!(  2,  9)(  3, 12)(  4, 20)(  5, 23)(  6, 13)(  7, 16)(  8, 24)
( 11, 25)( 15, 17)( 19, 21)( 26,151)( 27,159)( 28,162)( 29,170)( 30,173)
( 31,163)( 32,166)( 33,174)( 34,152)( 35,160)( 36,175)( 37,153)( 38,156)
( 39,164)( 40,167)( 41,157)( 42,165)( 43,168)( 44,171)( 45,154)( 46,169)
( 47,172)( 48,155)( 49,158)( 50,161)( 51, 76)( 52, 84)( 53, 87)( 54, 95)
( 55, 98)( 56, 88)( 57, 91)( 58, 99)( 59, 77)( 60, 85)( 61,100)( 62, 78)
( 63, 81)( 64, 89)( 65, 92)( 66, 82)( 67, 90)( 68, 93)( 69, 96)( 70, 79)
( 71, 94)( 72, 97)( 73, 80)( 74, 83)( 75, 86)(101,201)(102,209)(103,212)
(104,220)(105,223)(106,213)(107,216)(108,224)(109,202)(110,210)(111,225)
(112,203)(113,206)(114,214)(115,217)(116,207)(117,215)(118,218)(119,221)
(120,204)(121,219)(122,222)(123,205)(124,208)(125,211)(127,134)(128,137)
(129,145)(130,148)(131,138)(132,141)(133,149)(136,150)(140,142)(144,146)
(177,184)(178,187)(179,195)(180,198)(181,188)(182,191)(183,199)(186,200)
(190,192)(194,196);
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, s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*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*s0*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*s0*s1*s0*s1*s0*s1*s0*s1*s2*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