Polytope of Type {4,116}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,116}*1856
if this polytope has a name.
Group : SmallGroup(1856,737)
Rank : 3
Schlafli Type : {4,116}
Number of vertices, edges, etc : 8, 464, 232
Order of s0s1s2 : 116
Order of s0s1s2s1 : 4
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
None in this Atlas
Vertex Figure Of :
None in this Atlas
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {4,116}*928
4-fold quotients : {2,116}*464, {4,58}*464
8-fold quotients : {2,58}*232
16-fold quotients : {2,29}*116
29-fold quotients : {4,4}*64
58-fold quotients : {4,4}*32
116-fold quotients : {2,4}*16, {4,2}*16
232-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 59, 88)( 60, 89)( 61, 90)( 62, 91)( 63, 92)( 64, 93)( 65, 94)( 66, 95)
( 67, 96)( 68, 97)( 69, 98)( 70, 99)( 71,100)( 72,101)( 73,102)( 74,103)
( 75,104)( 76,105)( 77,106)( 78,107)( 79,108)( 80,109)( 81,110)( 82,111)
( 83,112)( 84,113)( 85,114)( 86,115)( 87,116)(175,204)(176,205)(177,206)
(178,207)(179,208)(180,209)(181,210)(182,211)(183,212)(184,213)(185,214)
(186,215)(187,216)(188,217)(189,218)(190,219)(191,220)(192,221)(193,222)
(194,223)(195,224)(196,225)(197,226)(198,227)(199,228)(200,229)(201,230)
(202,231)(203,232);;
s1 := ( 2, 29)( 3, 28)( 4, 27)( 5, 26)( 6, 25)( 7, 24)( 8, 23)( 9, 22)
( 10, 21)( 11, 20)( 12, 19)( 13, 18)( 14, 17)( 15, 16)( 31, 58)( 32, 57)
( 33, 56)( 34, 55)( 35, 54)( 36, 53)( 37, 52)( 38, 51)( 39, 50)( 40, 49)
( 41, 48)( 42, 47)( 43, 46)( 44, 45)( 60, 87)( 61, 86)( 62, 85)( 63, 84)
( 64, 83)( 65, 82)( 66, 81)( 67, 80)( 68, 79)( 69, 78)( 70, 77)( 71, 76)
( 72, 75)( 73, 74)( 89,116)( 90,115)( 91,114)( 92,113)( 93,112)( 94,111)
( 95,110)( 96,109)( 97,108)( 98,107)( 99,106)(100,105)(101,104)(102,103)
(117,175)(118,203)(119,202)(120,201)(121,200)(122,199)(123,198)(124,197)
(125,196)(126,195)(127,194)(128,193)(129,192)(130,191)(131,190)(132,189)
(133,188)(134,187)(135,186)(136,185)(137,184)(138,183)(139,182)(140,181)
(141,180)(142,179)(143,178)(144,177)(145,176)(146,204)(147,232)(148,231)
(149,230)(150,229)(151,228)(152,227)(153,226)(154,225)(155,224)(156,223)
(157,222)(158,221)(159,220)(160,219)(161,218)(162,217)(163,216)(164,215)
(165,214)(166,213)(167,212)(168,211)(169,210)(170,209)(171,208)(172,207)
(173,206)(174,205);;
s2 := ( 1,118)( 2,117)( 3,145)( 4,144)( 5,143)( 6,142)( 7,141)( 8,140)
( 9,139)( 10,138)( 11,137)( 12,136)( 13,135)( 14,134)( 15,133)( 16,132)
( 17,131)( 18,130)( 19,129)( 20,128)( 21,127)( 22,126)( 23,125)( 24,124)
( 25,123)( 26,122)( 27,121)( 28,120)( 29,119)( 30,147)( 31,146)( 32,174)
( 33,173)( 34,172)( 35,171)( 36,170)( 37,169)( 38,168)( 39,167)( 40,166)
( 41,165)( 42,164)( 43,163)( 44,162)( 45,161)( 46,160)( 47,159)( 48,158)
( 49,157)( 50,156)( 51,155)( 52,154)( 53,153)( 54,152)( 55,151)( 56,150)
( 57,149)( 58,148)( 59,176)( 60,175)( 61,203)( 62,202)( 63,201)( 64,200)
( 65,199)( 66,198)( 67,197)( 68,196)( 69,195)( 70,194)( 71,193)( 72,192)
( 73,191)( 74,190)( 75,189)( 76,188)( 77,187)( 78,186)( 79,185)( 80,184)
( 81,183)( 82,182)( 83,181)( 84,180)( 85,179)( 86,178)( 87,177)( 88,205)
( 89,204)( 90,232)( 91,231)( 92,230)( 93,229)( 94,228)( 95,227)( 96,226)
( 97,225)( 98,224)( 99,223)(100,222)(101,221)(102,220)(103,219)(104,218)
(105,217)(106,216)(107,215)(108,214)(109,213)(110,212)(111,211)(112,210)
(113,209)(114,208)(115,207)(116,206);;
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*s0*s1*s0*s1*s0*s1,
s2*s0*s1*s2*s1*s0*s1*s0*s2*s1*s2*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*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*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*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(232)!( 59, 88)( 60, 89)( 61, 90)( 62, 91)( 63, 92)( 64, 93)( 65, 94)
( 66, 95)( 67, 96)( 68, 97)( 69, 98)( 70, 99)( 71,100)( 72,101)( 73,102)
( 74,103)( 75,104)( 76,105)( 77,106)( 78,107)( 79,108)( 80,109)( 81,110)
( 82,111)( 83,112)( 84,113)( 85,114)( 86,115)( 87,116)(175,204)(176,205)
(177,206)(178,207)(179,208)(180,209)(181,210)(182,211)(183,212)(184,213)
(185,214)(186,215)(187,216)(188,217)(189,218)(190,219)(191,220)(192,221)
(193,222)(194,223)(195,224)(196,225)(197,226)(198,227)(199,228)(200,229)
(201,230)(202,231)(203,232);
s1 := Sym(232)!( 2, 29)( 3, 28)( 4, 27)( 5, 26)( 6, 25)( 7, 24)( 8, 23)
( 9, 22)( 10, 21)( 11, 20)( 12, 19)( 13, 18)( 14, 17)( 15, 16)( 31, 58)
( 32, 57)( 33, 56)( 34, 55)( 35, 54)( 36, 53)( 37, 52)( 38, 51)( 39, 50)
( 40, 49)( 41, 48)( 42, 47)( 43, 46)( 44, 45)( 60, 87)( 61, 86)( 62, 85)
( 63, 84)( 64, 83)( 65, 82)( 66, 81)( 67, 80)( 68, 79)( 69, 78)( 70, 77)
( 71, 76)( 72, 75)( 73, 74)( 89,116)( 90,115)( 91,114)( 92,113)( 93,112)
( 94,111)( 95,110)( 96,109)( 97,108)( 98,107)( 99,106)(100,105)(101,104)
(102,103)(117,175)(118,203)(119,202)(120,201)(121,200)(122,199)(123,198)
(124,197)(125,196)(126,195)(127,194)(128,193)(129,192)(130,191)(131,190)
(132,189)(133,188)(134,187)(135,186)(136,185)(137,184)(138,183)(139,182)
(140,181)(141,180)(142,179)(143,178)(144,177)(145,176)(146,204)(147,232)
(148,231)(149,230)(150,229)(151,228)(152,227)(153,226)(154,225)(155,224)
(156,223)(157,222)(158,221)(159,220)(160,219)(161,218)(162,217)(163,216)
(164,215)(165,214)(166,213)(167,212)(168,211)(169,210)(170,209)(171,208)
(172,207)(173,206)(174,205);
s2 := Sym(232)!( 1,118)( 2,117)( 3,145)( 4,144)( 5,143)( 6,142)( 7,141)
( 8,140)( 9,139)( 10,138)( 11,137)( 12,136)( 13,135)( 14,134)( 15,133)
( 16,132)( 17,131)( 18,130)( 19,129)( 20,128)( 21,127)( 22,126)( 23,125)
( 24,124)( 25,123)( 26,122)( 27,121)( 28,120)( 29,119)( 30,147)( 31,146)
( 32,174)( 33,173)( 34,172)( 35,171)( 36,170)( 37,169)( 38,168)( 39,167)
( 40,166)( 41,165)( 42,164)( 43,163)( 44,162)( 45,161)( 46,160)( 47,159)
( 48,158)( 49,157)( 50,156)( 51,155)( 52,154)( 53,153)( 54,152)( 55,151)
( 56,150)( 57,149)( 58,148)( 59,176)( 60,175)( 61,203)( 62,202)( 63,201)
( 64,200)( 65,199)( 66,198)( 67,197)( 68,196)( 69,195)( 70,194)( 71,193)
( 72,192)( 73,191)( 74,190)( 75,189)( 76,188)( 77,187)( 78,186)( 79,185)
( 80,184)( 81,183)( 82,182)( 83,181)( 84,180)( 85,179)( 86,178)( 87,177)
( 88,205)( 89,204)( 90,232)( 91,231)( 92,230)( 93,229)( 94,228)( 95,227)
( 96,226)( 97,225)( 98,224)( 99,223)(100,222)(101,221)(102,220)(103,219)
(104,218)(105,217)(106,216)(107,215)(108,214)(109,213)(110,212)(111,211)
(112,210)(113,209)(114,208)(115,207)(116,206);
poly := sub<Sym(232)|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*s0*s1*s0*s1*s0*s1,
s2*s0*s1*s2*s1*s0*s1*s0*s2*s1*s2*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*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*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*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