Polytope of Type {18,26}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {18,26}*936
Also Known As : {18,26|2}. if this polytope has another name.
Group : SmallGroup(936,64)
Rank : 3
Schlafli Type : {18,26}
Number of vertices, edges, etc : 18, 234, 26
Order of s0s1s2 : 234
Order of s0s1s2s1 : 2
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{18,26,2} of size 1872
Vertex Figure Of :
{2,18,26} of size 1872
Quotients (Maximal Quotients in Boldface) :
3-fold quotients : {6,26}*312
9-fold quotients : {2,26}*104
13-fold quotients : {18,2}*72
18-fold quotients : {2,13}*52
26-fold quotients : {9,2}*36
39-fold quotients : {6,2}*24
78-fold quotients : {3,2}*12
117-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {36,26}*1872, {18,52}*1872a
Permutation Representation (GAP) :
s0 := ( 2, 3)( 5, 6)( 8, 9)( 11, 12)( 14, 15)( 17, 18)( 20, 21)( 23, 24)
( 26, 27)( 29, 30)( 32, 33)( 35, 36)( 38, 39)( 40, 81)( 41, 80)( 42, 79)
( 43, 84)( 44, 83)( 45, 82)( 46, 87)( 47, 86)( 48, 85)( 49, 90)( 50, 89)
( 51, 88)( 52, 93)( 53, 92)( 54, 91)( 55, 96)( 56, 95)( 57, 94)( 58, 99)
( 59, 98)( 60, 97)( 61,102)( 62,101)( 63,100)( 64,105)( 65,104)( 66,103)
( 67,108)( 68,107)( 69,106)( 70,111)( 71,110)( 72,109)( 73,114)( 74,113)
( 75,112)( 76,117)( 77,116)( 78,115)(119,120)(122,123)(125,126)(128,129)
(131,132)(134,135)(137,138)(140,141)(143,144)(146,147)(149,150)(152,153)
(155,156)(157,198)(158,197)(159,196)(160,201)(161,200)(162,199)(163,204)
(164,203)(165,202)(166,207)(167,206)(168,205)(169,210)(170,209)(171,208)
(172,213)(173,212)(174,211)(175,216)(176,215)(177,214)(178,219)(179,218)
(180,217)(181,222)(182,221)(183,220)(184,225)(185,224)(186,223)(187,228)
(188,227)(189,226)(190,231)(191,230)(192,229)(193,234)(194,233)(195,232);;
s1 := ( 1, 40)( 2, 42)( 3, 41)( 4, 76)( 5, 78)( 6, 77)( 7, 73)( 8, 75)
( 9, 74)( 10, 70)( 11, 72)( 12, 71)( 13, 67)( 14, 69)( 15, 68)( 16, 64)
( 17, 66)( 18, 65)( 19, 61)( 20, 63)( 21, 62)( 22, 58)( 23, 60)( 24, 59)
( 25, 55)( 26, 57)( 27, 56)( 28, 52)( 29, 54)( 30, 53)( 31, 49)( 32, 51)
( 33, 50)( 34, 46)( 35, 48)( 36, 47)( 37, 43)( 38, 45)( 39, 44)( 79, 81)
( 82,117)( 83,116)( 84,115)( 85,114)( 86,113)( 87,112)( 88,111)( 89,110)
( 90,109)( 91,108)( 92,107)( 93,106)( 94,105)( 95,104)( 96,103)( 97,102)
( 98,101)( 99,100)(118,157)(119,159)(120,158)(121,193)(122,195)(123,194)
(124,190)(125,192)(126,191)(127,187)(128,189)(129,188)(130,184)(131,186)
(132,185)(133,181)(134,183)(135,182)(136,178)(137,180)(138,179)(139,175)
(140,177)(141,176)(142,172)(143,174)(144,173)(145,169)(146,171)(147,170)
(148,166)(149,168)(150,167)(151,163)(152,165)(153,164)(154,160)(155,162)
(156,161)(196,198)(199,234)(200,233)(201,232)(202,231)(203,230)(204,229)
(205,228)(206,227)(207,226)(208,225)(209,224)(210,223)(211,222)(212,221)
(213,220)(214,219)(215,218)(216,217);;
s2 := ( 1,121)( 2,122)( 3,123)( 4,118)( 5,119)( 6,120)( 7,154)( 8,155)
( 9,156)( 10,151)( 11,152)( 12,153)( 13,148)( 14,149)( 15,150)( 16,145)
( 17,146)( 18,147)( 19,142)( 20,143)( 21,144)( 22,139)( 23,140)( 24,141)
( 25,136)( 26,137)( 27,138)( 28,133)( 29,134)( 30,135)( 31,130)( 32,131)
( 33,132)( 34,127)( 35,128)( 36,129)( 37,124)( 38,125)( 39,126)( 40,160)
( 41,161)( 42,162)( 43,157)( 44,158)( 45,159)( 46,193)( 47,194)( 48,195)
( 49,190)( 50,191)( 51,192)( 52,187)( 53,188)( 54,189)( 55,184)( 56,185)
( 57,186)( 58,181)( 59,182)( 60,183)( 61,178)( 62,179)( 63,180)( 64,175)
( 65,176)( 66,177)( 67,172)( 68,173)( 69,174)( 70,169)( 71,170)( 72,171)
( 73,166)( 74,167)( 75,168)( 76,163)( 77,164)( 78,165)( 79,199)( 80,200)
( 81,201)( 82,196)( 83,197)( 84,198)( 85,232)( 86,233)( 87,234)( 88,229)
( 89,230)( 90,231)( 91,226)( 92,227)( 93,228)( 94,223)( 95,224)( 96,225)
( 97,220)( 98,221)( 99,222)(100,217)(101,218)(102,219)(103,214)(104,215)
(105,216)(106,211)(107,212)(108,213)(109,208)(110,209)(111,210)(112,205)
(113,206)(114,207)(115,202)(116,203)(117,204);;
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*s2*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*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 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(234)!( 2, 3)( 5, 6)( 8, 9)( 11, 12)( 14, 15)( 17, 18)( 20, 21)
( 23, 24)( 26, 27)( 29, 30)( 32, 33)( 35, 36)( 38, 39)( 40, 81)( 41, 80)
( 42, 79)( 43, 84)( 44, 83)( 45, 82)( 46, 87)( 47, 86)( 48, 85)( 49, 90)
( 50, 89)( 51, 88)( 52, 93)( 53, 92)( 54, 91)( 55, 96)( 56, 95)( 57, 94)
( 58, 99)( 59, 98)( 60, 97)( 61,102)( 62,101)( 63,100)( 64,105)( 65,104)
( 66,103)( 67,108)( 68,107)( 69,106)( 70,111)( 71,110)( 72,109)( 73,114)
( 74,113)( 75,112)( 76,117)( 77,116)( 78,115)(119,120)(122,123)(125,126)
(128,129)(131,132)(134,135)(137,138)(140,141)(143,144)(146,147)(149,150)
(152,153)(155,156)(157,198)(158,197)(159,196)(160,201)(161,200)(162,199)
(163,204)(164,203)(165,202)(166,207)(167,206)(168,205)(169,210)(170,209)
(171,208)(172,213)(173,212)(174,211)(175,216)(176,215)(177,214)(178,219)
(179,218)(180,217)(181,222)(182,221)(183,220)(184,225)(185,224)(186,223)
(187,228)(188,227)(189,226)(190,231)(191,230)(192,229)(193,234)(194,233)
(195,232);
s1 := Sym(234)!( 1, 40)( 2, 42)( 3, 41)( 4, 76)( 5, 78)( 6, 77)( 7, 73)
( 8, 75)( 9, 74)( 10, 70)( 11, 72)( 12, 71)( 13, 67)( 14, 69)( 15, 68)
( 16, 64)( 17, 66)( 18, 65)( 19, 61)( 20, 63)( 21, 62)( 22, 58)( 23, 60)
( 24, 59)( 25, 55)( 26, 57)( 27, 56)( 28, 52)( 29, 54)( 30, 53)( 31, 49)
( 32, 51)( 33, 50)( 34, 46)( 35, 48)( 36, 47)( 37, 43)( 38, 45)( 39, 44)
( 79, 81)( 82,117)( 83,116)( 84,115)( 85,114)( 86,113)( 87,112)( 88,111)
( 89,110)( 90,109)( 91,108)( 92,107)( 93,106)( 94,105)( 95,104)( 96,103)
( 97,102)( 98,101)( 99,100)(118,157)(119,159)(120,158)(121,193)(122,195)
(123,194)(124,190)(125,192)(126,191)(127,187)(128,189)(129,188)(130,184)
(131,186)(132,185)(133,181)(134,183)(135,182)(136,178)(137,180)(138,179)
(139,175)(140,177)(141,176)(142,172)(143,174)(144,173)(145,169)(146,171)
(147,170)(148,166)(149,168)(150,167)(151,163)(152,165)(153,164)(154,160)
(155,162)(156,161)(196,198)(199,234)(200,233)(201,232)(202,231)(203,230)
(204,229)(205,228)(206,227)(207,226)(208,225)(209,224)(210,223)(211,222)
(212,221)(213,220)(214,219)(215,218)(216,217);
s2 := Sym(234)!( 1,121)( 2,122)( 3,123)( 4,118)( 5,119)( 6,120)( 7,154)
( 8,155)( 9,156)( 10,151)( 11,152)( 12,153)( 13,148)( 14,149)( 15,150)
( 16,145)( 17,146)( 18,147)( 19,142)( 20,143)( 21,144)( 22,139)( 23,140)
( 24,141)( 25,136)( 26,137)( 27,138)( 28,133)( 29,134)( 30,135)( 31,130)
( 32,131)( 33,132)( 34,127)( 35,128)( 36,129)( 37,124)( 38,125)( 39,126)
( 40,160)( 41,161)( 42,162)( 43,157)( 44,158)( 45,159)( 46,193)( 47,194)
( 48,195)( 49,190)( 50,191)( 51,192)( 52,187)( 53,188)( 54,189)( 55,184)
( 56,185)( 57,186)( 58,181)( 59,182)( 60,183)( 61,178)( 62,179)( 63,180)
( 64,175)( 65,176)( 66,177)( 67,172)( 68,173)( 69,174)( 70,169)( 71,170)
( 72,171)( 73,166)( 74,167)( 75,168)( 76,163)( 77,164)( 78,165)( 79,199)
( 80,200)( 81,201)( 82,196)( 83,197)( 84,198)( 85,232)( 86,233)( 87,234)
( 88,229)( 89,230)( 90,231)( 91,226)( 92,227)( 93,228)( 94,223)( 95,224)
( 96,225)( 97,220)( 98,221)( 99,222)(100,217)(101,218)(102,219)(103,214)
(104,215)(105,216)(106,211)(107,212)(108,213)(109,208)(110,209)(111,210)
(112,205)(113,206)(114,207)(115,202)(116,203)(117,204);
poly := sub<Sym(234)|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*s2*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*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 >;
References : None.
to this polytope