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