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Polytope of Type {4,38,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,38,6}*1824
Also Known As : {{4,38|2},{38,6|2}}. if this polytope has another name.
Group : SmallGroup(1824,1141)
Rank : 4
Schlafli Type : {4,38,6}
Number of vertices, edges, etc : 4, 76, 114, 6
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 : {2,38,6}*912
   3-fold quotients : {4,38,2}*608
   6-fold quotients : {2,38,2}*304
   12-fold quotients : {2,19,2}*152
   19-fold quotients : {4,2,6}*96
   38-fold quotients : {4,2,3}*48, {2,2,6}*48
   57-fold quotients : {4,2,2}*32
   76-fold quotients : {2,2,3}*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)(  2,133)(  3,132)(  4,131)(  5,130)(  6,129)(  7,128)(  8,127)
(  9,126)( 10,125)( 11,124)( 12,123)( 13,122)( 14,121)( 15,120)( 16,119)
( 17,118)( 18,117)( 19,116)( 20,134)( 21,152)( 22,151)( 23,150)( 24,149)
( 25,148)( 26,147)( 27,146)( 28,145)( 29,144)( 30,143)( 31,142)( 32,141)
( 33,140)( 34,139)( 35,138)( 36,137)( 37,136)( 38,135)( 39,153)( 40,171)
( 41,170)( 42,169)( 43,168)( 44,167)( 45,166)( 46,165)( 47,164)( 48,163)
( 49,162)( 50,161)( 51,160)( 52,159)( 53,158)( 54,157)( 55,156)( 56,155)
( 57,154)( 58,172)( 59,190)( 60,189)( 61,188)( 62,187)( 63,186)( 64,185)
( 65,184)( 66,183)( 67,182)( 68,181)( 69,180)( 70,179)( 71,178)( 72,177)
( 73,176)( 74,175)( 75,174)( 76,173)( 77,191)( 78,209)( 79,208)( 80,207)
( 81,206)( 82,205)( 83,204)( 84,203)( 85,202)( 86,201)( 87,200)( 88,199)
( 89,198)( 90,197)( 91,196)( 92,195)( 93,194)( 94,193)( 95,192)( 96,210)
( 97,228)( 98,227)( 99,226)(100,225)(101,224)(102,223)(103,222)(104,221)
(105,220)(106,219)(107,218)(108,217)(109,216)(110,215)(111,214)(112,213)
(113,212)(114,211);;
s2 := (  1,  2)(  3, 19)(  4, 18)(  5, 17)(  6, 16)(  7, 15)(  8, 14)(  9, 13)
( 10, 12)( 20, 40)( 21, 39)( 22, 57)( 23, 56)( 24, 55)( 25, 54)( 26, 53)
( 27, 52)( 28, 51)( 29, 50)( 30, 49)( 31, 48)( 32, 47)( 33, 46)( 34, 45)
( 35, 44)( 36, 43)( 37, 42)( 38, 41)( 58, 59)( 60, 76)( 61, 75)( 62, 74)
( 63, 73)( 64, 72)( 65, 71)( 66, 70)( 67, 69)( 77, 97)( 78, 96)( 79,114)
( 80,113)( 81,112)( 82,111)( 83,110)( 84,109)( 85,108)( 86,107)( 87,106)
( 88,105)( 89,104)( 90,103)( 91,102)( 92,101)( 93,100)( 94, 99)( 95, 98)
(115,116)(117,133)(118,132)(119,131)(120,130)(121,129)(122,128)(123,127)
(124,126)(134,154)(135,153)(136,171)(137,170)(138,169)(139,168)(140,167)
(141,166)(142,165)(143,164)(144,163)(145,162)(146,161)(147,160)(148,159)
(149,158)(150,157)(151,156)(152,155)(172,173)(174,190)(175,189)(176,188)
(177,187)(178,186)(179,185)(180,184)(181,183)(191,211)(192,210)(193,228)
(194,227)(195,226)(196,225)(197,224)(198,223)(199,222)(200,221)(201,220)
(202,219)(203,218)(204,217)(205,216)(206,215)(207,214)(208,213)(209,212);;
s3 := (  1, 20)(  2, 21)(  3, 22)(  4, 23)(  5, 24)(  6, 25)(  7, 26)(  8, 27)
(  9, 28)( 10, 29)( 11, 30)( 12, 31)( 13, 32)( 14, 33)( 15, 34)( 16, 35)
( 17, 36)( 18, 37)( 19, 38)( 58, 77)( 59, 78)( 60, 79)( 61, 80)( 62, 81)
( 63, 82)( 64, 83)( 65, 84)( 66, 85)( 67, 86)( 68, 87)( 69, 88)( 70, 89)
( 71, 90)( 72, 91)( 73, 92)( 74, 93)( 75, 94)( 76, 95)(115,134)(116,135)
(117,136)(118,137)(119,138)(120,139)(121,140)(122,141)(123,142)(124,143)
(125,144)(126,145)(127,146)(128,147)(129,148)(130,149)(131,150)(132,151)
(133,152)(172,191)(173,192)(174,193)(175,194)(176,195)(177,196)(178,197)
(179,198)(180,199)(181,200)(182,201)(183,202)(184,203)(185,204)(186,205)
(187,206)(188,207)(189,208)(190,209);;
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, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*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)!(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)(  2,133)(  3,132)(  4,131)(  5,130)(  6,129)(  7,128)
(  8,127)(  9,126)( 10,125)( 11,124)( 12,123)( 13,122)( 14,121)( 15,120)
( 16,119)( 17,118)( 18,117)( 19,116)( 20,134)( 21,152)( 22,151)( 23,150)
( 24,149)( 25,148)( 26,147)( 27,146)( 28,145)( 29,144)( 30,143)( 31,142)
( 32,141)( 33,140)( 34,139)( 35,138)( 36,137)( 37,136)( 38,135)( 39,153)
( 40,171)( 41,170)( 42,169)( 43,168)( 44,167)( 45,166)( 46,165)( 47,164)
( 48,163)( 49,162)( 50,161)( 51,160)( 52,159)( 53,158)( 54,157)( 55,156)
( 56,155)( 57,154)( 58,172)( 59,190)( 60,189)( 61,188)( 62,187)( 63,186)
( 64,185)( 65,184)( 66,183)( 67,182)( 68,181)( 69,180)( 70,179)( 71,178)
( 72,177)( 73,176)( 74,175)( 75,174)( 76,173)( 77,191)( 78,209)( 79,208)
( 80,207)( 81,206)( 82,205)( 83,204)( 84,203)( 85,202)( 86,201)( 87,200)
( 88,199)( 89,198)( 90,197)( 91,196)( 92,195)( 93,194)( 94,193)( 95,192)
( 96,210)( 97,228)( 98,227)( 99,226)(100,225)(101,224)(102,223)(103,222)
(104,221)(105,220)(106,219)(107,218)(108,217)(109,216)(110,215)(111,214)
(112,213)(113,212)(114,211);
s2 := Sym(228)!(  1,  2)(  3, 19)(  4, 18)(  5, 17)(  6, 16)(  7, 15)(  8, 14)
(  9, 13)( 10, 12)( 20, 40)( 21, 39)( 22, 57)( 23, 56)( 24, 55)( 25, 54)
( 26, 53)( 27, 52)( 28, 51)( 29, 50)( 30, 49)( 31, 48)( 32, 47)( 33, 46)
( 34, 45)( 35, 44)( 36, 43)( 37, 42)( 38, 41)( 58, 59)( 60, 76)( 61, 75)
( 62, 74)( 63, 73)( 64, 72)( 65, 71)( 66, 70)( 67, 69)( 77, 97)( 78, 96)
( 79,114)( 80,113)( 81,112)( 82,111)( 83,110)( 84,109)( 85,108)( 86,107)
( 87,106)( 88,105)( 89,104)( 90,103)( 91,102)( 92,101)( 93,100)( 94, 99)
( 95, 98)(115,116)(117,133)(118,132)(119,131)(120,130)(121,129)(122,128)
(123,127)(124,126)(134,154)(135,153)(136,171)(137,170)(138,169)(139,168)
(140,167)(141,166)(142,165)(143,164)(144,163)(145,162)(146,161)(147,160)
(148,159)(149,158)(150,157)(151,156)(152,155)(172,173)(174,190)(175,189)
(176,188)(177,187)(178,186)(179,185)(180,184)(181,183)(191,211)(192,210)
(193,228)(194,227)(195,226)(196,225)(197,224)(198,223)(199,222)(200,221)
(201,220)(202,219)(203,218)(204,217)(205,216)(206,215)(207,214)(208,213)
(209,212);
s3 := Sym(228)!(  1, 20)(  2, 21)(  3, 22)(  4, 23)(  5, 24)(  6, 25)(  7, 26)
(  8, 27)(  9, 28)( 10, 29)( 11, 30)( 12, 31)( 13, 32)( 14, 33)( 15, 34)
( 16, 35)( 17, 36)( 18, 37)( 19, 38)( 58, 77)( 59, 78)( 60, 79)( 61, 80)
( 62, 81)( 63, 82)( 64, 83)( 65, 84)( 66, 85)( 67, 86)( 68, 87)( 69, 88)
( 70, 89)( 71, 90)( 72, 91)( 73, 92)( 74, 93)( 75, 94)( 76, 95)(115,134)
(116,135)(117,136)(118,137)(119,138)(120,139)(121,140)(122,141)(123,142)
(124,143)(125,144)(126,145)(127,146)(128,147)(129,148)(130,149)(131,150)
(132,151)(133,152)(172,191)(173,192)(174,193)(175,194)(176,195)(177,196)
(178,197)(179,198)(180,199)(181,200)(182,201)(183,202)(184,203)(185,204)
(186,205)(187,206)(188,207)(189,208)(190,209);
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, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*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