Polytope of Type {6,38,4}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,38,4}*1824
Also Known As : {{6,38|2},{38,4|2}}. if this polytope has another name.
Group : SmallGroup(1824,1141)
Rank : 4
Schlafli Type : {6,38,4}
Number of vertices, edges, etc : 6, 114, 76, 4
Order of s0s1s2s3 : 228
Order of s0s1s2s3s2s1 : 2
Special Properties :
Universal
Orientable
Flat
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 : {6,38,2}*912
3-fold quotients : {2,38,4}*608
6-fold quotients : {2,38,2}*304
12-fold quotients : {2,19,2}*152
19-fold quotients : {6,2,4}*96
38-fold quotients : {3,2,4}*48, {6,2,2}*48
57-fold quotients : {2,2,4}*32
76-fold quotients : {3,2,2}*24
114-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 20, 39)( 21, 40)( 22, 41)( 23, 42)( 24, 43)( 25, 44)( 26, 45)( 27, 46)
( 28, 47)( 29, 48)( 30, 49)( 31, 50)( 32, 51)( 33, 52)( 34, 53)( 35, 54)
( 36, 55)( 37, 56)( 38, 57)( 77, 96)( 78, 97)( 79, 98)( 80, 99)( 81,100)
( 82,101)( 83,102)( 84,103)( 85,104)( 86,105)( 87,106)( 88,107)( 89,108)
( 90,109)( 91,110)( 92,111)( 93,112)( 94,113)( 95,114)(134,153)(135,154)
(136,155)(137,156)(138,157)(139,158)(140,159)(141,160)(142,161)(143,162)
(144,163)(145,164)(146,165)(147,166)(148,167)(149,168)(150,169)(151,170)
(152,171)(191,210)(192,211)(193,212)(194,213)(195,214)(196,215)(197,216)
(198,217)(199,218)(200,219)(201,220)(202,221)(203,222)(204,223)(205,224)
(206,225)(207,226)(208,227)(209,228);;
s1 := ( 1, 20)( 2, 38)( 3, 37)( 4, 36)( 5, 35)( 6, 34)( 7, 33)( 8, 32)
( 9, 31)( 10, 30)( 11, 29)( 12, 28)( 13, 27)( 14, 26)( 15, 25)( 16, 24)
( 17, 23)( 18, 22)( 19, 21)( 40, 57)( 41, 56)( 42, 55)( 43, 54)( 44, 53)
( 45, 52)( 46, 51)( 47, 50)( 48, 49)( 58, 77)( 59, 95)( 60, 94)( 61, 93)
( 62, 92)( 63, 91)( 64, 90)( 65, 89)( 66, 88)( 67, 87)( 68, 86)( 69, 85)
( 70, 84)( 71, 83)( 72, 82)( 73, 81)( 74, 80)( 75, 79)( 76, 78)( 97,114)
( 98,113)( 99,112)(100,111)(101,110)(102,109)(103,108)(104,107)(105,106)
(115,134)(116,152)(117,151)(118,150)(119,149)(120,148)(121,147)(122,146)
(123,145)(124,144)(125,143)(126,142)(127,141)(128,140)(129,139)(130,138)
(131,137)(132,136)(133,135)(154,171)(155,170)(156,169)(157,168)(158,167)
(159,166)(160,165)(161,164)(162,163)(172,191)(173,209)(174,208)(175,207)
(176,206)(177,205)(178,204)(179,203)(180,202)(181,201)(182,200)(183,199)
(184,198)(185,197)(186,196)(187,195)(188,194)(189,193)(190,192)(211,228)
(212,227)(213,226)(214,225)(215,224)(216,223)(217,222)(218,221)(219,220);;
s2 := ( 1, 2)( 3, 19)( 4, 18)( 5, 17)( 6, 16)( 7, 15)( 8, 14)( 9, 13)
( 10, 12)( 20, 21)( 22, 38)( 23, 37)( 24, 36)( 25, 35)( 26, 34)( 27, 33)
( 28, 32)( 29, 31)( 39, 40)( 41, 57)( 42, 56)( 43, 55)( 44, 54)( 45, 53)
( 46, 52)( 47, 51)( 48, 50)( 58, 59)( 60, 76)( 61, 75)( 62, 74)( 63, 73)
( 64, 72)( 65, 71)( 66, 70)( 67, 69)( 77, 78)( 79, 95)( 80, 94)( 81, 93)
( 82, 92)( 83, 91)( 84, 90)( 85, 89)( 86, 88)( 96, 97)( 98,114)( 99,113)
(100,112)(101,111)(102,110)(103,109)(104,108)(105,107)(115,173)(116,172)
(117,190)(118,189)(119,188)(120,187)(121,186)(122,185)(123,184)(124,183)
(125,182)(126,181)(127,180)(128,179)(129,178)(130,177)(131,176)(132,175)
(133,174)(134,192)(135,191)(136,209)(137,208)(138,207)(139,206)(140,205)
(141,204)(142,203)(143,202)(144,201)(145,200)(146,199)(147,198)(148,197)
(149,196)(150,195)(151,194)(152,193)(153,211)(154,210)(155,228)(156,227)
(157,226)(158,225)(159,224)(160,223)(161,222)(162,221)(163,220)(164,219)
(165,218)(166,217)(167,216)(168,215)(169,214)(170,213)(171,212);;
s3 := ( 1,115)( 2,116)( 3,117)( 4,118)( 5,119)( 6,120)( 7,121)( 8,122)
( 9,123)( 10,124)( 11,125)( 12,126)( 13,127)( 14,128)( 15,129)( 16,130)
( 17,131)( 18,132)( 19,133)( 20,134)( 21,135)( 22,136)( 23,137)( 24,138)
( 25,139)( 26,140)( 27,141)( 28,142)( 29,143)( 30,144)( 31,145)( 32,146)
( 33,147)( 34,148)( 35,149)( 36,150)( 37,151)( 38,152)( 39,153)( 40,154)
( 41,155)( 42,156)( 43,157)( 44,158)( 45,159)( 46,160)( 47,161)( 48,162)
( 49,163)( 50,164)( 51,165)( 52,166)( 53,167)( 54,168)( 55,169)( 56,170)
( 57,171)( 58,172)( 59,173)( 60,174)( 61,175)( 62,176)( 63,177)( 64,178)
( 65,179)( 66,180)( 67,181)( 68,182)( 69,183)( 70,184)( 71,185)( 72,186)
( 73,187)( 74,188)( 75,189)( 76,190)( 77,191)( 78,192)( 79,193)( 80,194)
( 81,195)( 82,196)( 83,197)( 84,198)( 85,199)( 86,200)( 87,201)( 88,202)
( 89,203)( 90,204)( 91,205)( 92,206)( 93,207)( 94,208)( 95,209)( 96,210)
( 97,211)( 98,212)( 99,213)(100,214)(101,215)(102,216)(103,217)(104,218)
(105,219)(106,220)(107,221)(108,222)(109,223)(110,224)(111,225)(112,226)
(113,227)(114,228);;
poly := Group([s0,s1,s2,s3]);;
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;; s1 := F.2;; s2 := F.3;; s3 := F.4;;
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s2*s0*s2,
s0*s3*s0*s3, s1*s3*s1*s3, s0*s1*s2*s1*s0*s1*s2*s1,
s1*s2*s3*s2*s1*s2*s3*s2, s2*s3*s2*s3*s2*s3*s2*s3,
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*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(228)!( 20, 39)( 21, 40)( 22, 41)( 23, 42)( 24, 43)( 25, 44)( 26, 45)
( 27, 46)( 28, 47)( 29, 48)( 30, 49)( 31, 50)( 32, 51)( 33, 52)( 34, 53)
( 35, 54)( 36, 55)( 37, 56)( 38, 57)( 77, 96)( 78, 97)( 79, 98)( 80, 99)
( 81,100)( 82,101)( 83,102)( 84,103)( 85,104)( 86,105)( 87,106)( 88,107)
( 89,108)( 90,109)( 91,110)( 92,111)( 93,112)( 94,113)( 95,114)(134,153)
(135,154)(136,155)(137,156)(138,157)(139,158)(140,159)(141,160)(142,161)
(143,162)(144,163)(145,164)(146,165)(147,166)(148,167)(149,168)(150,169)
(151,170)(152,171)(191,210)(192,211)(193,212)(194,213)(195,214)(196,215)
(197,216)(198,217)(199,218)(200,219)(201,220)(202,221)(203,222)(204,223)
(205,224)(206,225)(207,226)(208,227)(209,228);
s1 := Sym(228)!( 1, 20)( 2, 38)( 3, 37)( 4, 36)( 5, 35)( 6, 34)( 7, 33)
( 8, 32)( 9, 31)( 10, 30)( 11, 29)( 12, 28)( 13, 27)( 14, 26)( 15, 25)
( 16, 24)( 17, 23)( 18, 22)( 19, 21)( 40, 57)( 41, 56)( 42, 55)( 43, 54)
( 44, 53)( 45, 52)( 46, 51)( 47, 50)( 48, 49)( 58, 77)( 59, 95)( 60, 94)
( 61, 93)( 62, 92)( 63, 91)( 64, 90)( 65, 89)( 66, 88)( 67, 87)( 68, 86)
( 69, 85)( 70, 84)( 71, 83)( 72, 82)( 73, 81)( 74, 80)( 75, 79)( 76, 78)
( 97,114)( 98,113)( 99,112)(100,111)(101,110)(102,109)(103,108)(104,107)
(105,106)(115,134)(116,152)(117,151)(118,150)(119,149)(120,148)(121,147)
(122,146)(123,145)(124,144)(125,143)(126,142)(127,141)(128,140)(129,139)
(130,138)(131,137)(132,136)(133,135)(154,171)(155,170)(156,169)(157,168)
(158,167)(159,166)(160,165)(161,164)(162,163)(172,191)(173,209)(174,208)
(175,207)(176,206)(177,205)(178,204)(179,203)(180,202)(181,201)(182,200)
(183,199)(184,198)(185,197)(186,196)(187,195)(188,194)(189,193)(190,192)
(211,228)(212,227)(213,226)(214,225)(215,224)(216,223)(217,222)(218,221)
(219,220);
s2 := Sym(228)!( 1, 2)( 3, 19)( 4, 18)( 5, 17)( 6, 16)( 7, 15)( 8, 14)
( 9, 13)( 10, 12)( 20, 21)( 22, 38)( 23, 37)( 24, 36)( 25, 35)( 26, 34)
( 27, 33)( 28, 32)( 29, 31)( 39, 40)( 41, 57)( 42, 56)( 43, 55)( 44, 54)
( 45, 53)( 46, 52)( 47, 51)( 48, 50)( 58, 59)( 60, 76)( 61, 75)( 62, 74)
( 63, 73)( 64, 72)( 65, 71)( 66, 70)( 67, 69)( 77, 78)( 79, 95)( 80, 94)
( 81, 93)( 82, 92)( 83, 91)( 84, 90)( 85, 89)( 86, 88)( 96, 97)( 98,114)
( 99,113)(100,112)(101,111)(102,110)(103,109)(104,108)(105,107)(115,173)
(116,172)(117,190)(118,189)(119,188)(120,187)(121,186)(122,185)(123,184)
(124,183)(125,182)(126,181)(127,180)(128,179)(129,178)(130,177)(131,176)
(132,175)(133,174)(134,192)(135,191)(136,209)(137,208)(138,207)(139,206)
(140,205)(141,204)(142,203)(143,202)(144,201)(145,200)(146,199)(147,198)
(148,197)(149,196)(150,195)(151,194)(152,193)(153,211)(154,210)(155,228)
(156,227)(157,226)(158,225)(159,224)(160,223)(161,222)(162,221)(163,220)
(164,219)(165,218)(166,217)(167,216)(168,215)(169,214)(170,213)(171,212);
s3 := Sym(228)!( 1,115)( 2,116)( 3,117)( 4,118)( 5,119)( 6,120)( 7,121)
( 8,122)( 9,123)( 10,124)( 11,125)( 12,126)( 13,127)( 14,128)( 15,129)
( 16,130)( 17,131)( 18,132)( 19,133)( 20,134)( 21,135)( 22,136)( 23,137)
( 24,138)( 25,139)( 26,140)( 27,141)( 28,142)( 29,143)( 30,144)( 31,145)
( 32,146)( 33,147)( 34,148)( 35,149)( 36,150)( 37,151)( 38,152)( 39,153)
( 40,154)( 41,155)( 42,156)( 43,157)( 44,158)( 45,159)( 46,160)( 47,161)
( 48,162)( 49,163)( 50,164)( 51,165)( 52,166)( 53,167)( 54,168)( 55,169)
( 56,170)( 57,171)( 58,172)( 59,173)( 60,174)( 61,175)( 62,176)( 63,177)
( 64,178)( 65,179)( 66,180)( 67,181)( 68,182)( 69,183)( 70,184)( 71,185)
( 72,186)( 73,187)( 74,188)( 75,189)( 76,190)( 77,191)( 78,192)( 79,193)
( 80,194)( 81,195)( 82,196)( 83,197)( 84,198)( 85,199)( 86,200)( 87,201)
( 88,202)( 89,203)( 90,204)( 91,205)( 92,206)( 93,207)( 94,208)( 95,209)
( 96,210)( 97,211)( 98,212)( 99,213)(100,214)(101,215)(102,216)(103,217)
(104,218)(105,219)(106,220)(107,221)(108,222)(109,223)(110,224)(111,225)
(112,226)(113,227)(114,228);
poly := sub<Sym(228)|s0,s1,s2,s3>;
Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2,
s3*s3, s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3,
s0*s1*s2*s1*s0*s1*s2*s1, s1*s2*s3*s2*s1*s2*s3*s2,
s2*s3*s2*s3*s2*s3*s2*s3, 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*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;
References : None.
to this polytope