Polytope of Type {76,6,2}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {76,6,2}*1824a
if this polytope has a name.
Group : SmallGroup(1824,1132)
Rank : 4
Schlafli Type : {76,6,2}
Number of vertices, edges, etc : 76, 228, 6, 2
Order of s₀s₁s₂s₃ : 228
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Degenerate
   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 : {38,6,2}*912
   3-fold quotients : {76,2,2}*608
   6-fold quotients : {38,2,2}*304
   12-fold quotients : {19,2,2}*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 := (  2, 19)(  3, 18)(  4, 17)(  5, 16)(  6, 15)(  7, 14)(  8, 13)(  9, 12)
( 10, 11)( 21, 38)( 22, 37)( 23, 36)( 24, 35)( 25, 34)( 26, 33)( 27, 32)
( 28, 31)( 29, 30)( 40, 57)( 41, 56)( 42, 55)( 43, 54)( 44, 53)( 45, 52)
( 46, 51)( 47, 50)( 48, 49)( 59, 76)( 60, 75)( 61, 74)( 62, 73)( 63, 72)
( 64, 71)( 65, 70)( 66, 69)( 67, 68)( 78, 95)( 79, 94)( 80, 93)( 81, 92)
( 82, 91)( 83, 90)( 84, 89)( 85, 88)( 86, 87)( 97,114)( 98,113)( 99,112)
(100,111)(101,110)(102,109)(103,108)(104,107)(105,106)(115,172)(116,190)
(117,189)(118,188)(119,187)(120,186)(121,185)(122,184)(123,183)(124,182)
(125,181)(126,180)(127,179)(128,178)(129,177)(130,176)(131,175)(132,174)
(133,173)(134,191)(135,209)(136,208)(137,207)(138,206)(139,205)(140,204)
(141,203)(142,202)(143,201)(144,200)(145,199)(146,198)(147,197)(148,196)
(149,195)(150,194)(151,193)(152,192)(153,210)(154,228)(155,227)(156,226)
(157,225)(158,224)(159,223)(160,222)(161,221)(162,220)(163,219)(164,218)
(165,217)(166,216)(167,215)(168,214)(169,213)(170,212)(171,211);;
s1 := (  1,116)(  2,115)(  3,133)(  4,132)(  5,131)(  6,130)(  7,129)(  8,128)
(  9,127)( 10,126)( 11,125)( 12,124)( 13,123)( 14,122)( 15,121)( 16,120)
( 17,119)( 18,118)( 19,117)( 20,154)( 21,153)( 22,171)( 23,170)( 24,169)
( 25,168)( 26,167)( 27,166)( 28,165)( 29,164)( 30,163)( 31,162)( 32,161)
( 33,160)( 34,159)( 35,158)( 36,157)( 37,156)( 38,155)( 39,135)( 40,134)
( 41,152)( 42,151)( 43,150)( 44,149)( 45,148)( 46,147)( 47,146)( 48,145)
( 49,144)( 50,143)( 51,142)( 52,141)( 53,140)( 54,139)( 55,138)( 56,137)
( 57,136)( 58,173)( 59,172)( 60,190)( 61,189)( 62,188)( 63,187)( 64,186)
( 65,185)( 66,184)( 67,183)( 68,182)( 69,181)( 70,180)( 71,179)( 72,178)
( 73,177)( 74,176)( 75,175)( 76,174)( 77,211)( 78,210)( 79,228)( 80,227)
( 81,226)( 82,225)( 83,224)( 84,223)( 85,222)( 86,221)( 87,220)( 88,219)
( 89,218)( 90,217)( 91,216)( 92,215)( 93,214)( 94,213)( 95,212)( 96,192)
( 97,191)( 98,209)( 99,208)(100,207)(101,206)(102,205)(103,204)(104,203)
(105,202)(106,201)(107,200)(108,199)(109,198)(110,197)(111,196)(112,195)
(113,194)(114,193);;
s2 := (  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);;
s3 := (229,230);;
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, s2*s3*s2*s3, 
s0*s1*s2*s1*s0*s1*s2*s1, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
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*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*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*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*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(230)!(  2, 19)(  3, 18)(  4, 17)(  5, 16)(  6, 15)(  7, 14)(  8, 13)
(  9, 12)( 10, 11)( 21, 38)( 22, 37)( 23, 36)( 24, 35)( 25, 34)( 26, 33)
( 27, 32)( 28, 31)( 29, 30)( 40, 57)( 41, 56)( 42, 55)( 43, 54)( 44, 53)
( 45, 52)( 46, 51)( 47, 50)( 48, 49)( 59, 76)( 60, 75)( 61, 74)( 62, 73)
( 63, 72)( 64, 71)( 65, 70)( 66, 69)( 67, 68)( 78, 95)( 79, 94)( 80, 93)
( 81, 92)( 82, 91)( 83, 90)( 84, 89)( 85, 88)( 86, 87)( 97,114)( 98,113)
( 99,112)(100,111)(101,110)(102,109)(103,108)(104,107)(105,106)(115,172)
(116,190)(117,189)(118,188)(119,187)(120,186)(121,185)(122,184)(123,183)
(124,182)(125,181)(126,180)(127,179)(128,178)(129,177)(130,176)(131,175)
(132,174)(133,173)(134,191)(135,209)(136,208)(137,207)(138,206)(139,205)
(140,204)(141,203)(142,202)(143,201)(144,200)(145,199)(146,198)(147,197)
(148,196)(149,195)(150,194)(151,193)(152,192)(153,210)(154,228)(155,227)
(156,226)(157,225)(158,224)(159,223)(160,222)(161,221)(162,220)(163,219)
(164,218)(165,217)(166,216)(167,215)(168,214)(169,213)(170,212)(171,211);
s1 := Sym(230)!(  1,116)(  2,115)(  3,133)(  4,132)(  5,131)(  6,130)(  7,129)
(  8,128)(  9,127)( 10,126)( 11,125)( 12,124)( 13,123)( 14,122)( 15,121)
( 16,120)( 17,119)( 18,118)( 19,117)( 20,154)( 21,153)( 22,171)( 23,170)
( 24,169)( 25,168)( 26,167)( 27,166)( 28,165)( 29,164)( 30,163)( 31,162)
( 32,161)( 33,160)( 34,159)( 35,158)( 36,157)( 37,156)( 38,155)( 39,135)
( 40,134)( 41,152)( 42,151)( 43,150)( 44,149)( 45,148)( 46,147)( 47,146)
( 48,145)( 49,144)( 50,143)( 51,142)( 52,141)( 53,140)( 54,139)( 55,138)
( 56,137)( 57,136)( 58,173)( 59,172)( 60,190)( 61,189)( 62,188)( 63,187)
( 64,186)( 65,185)( 66,184)( 67,183)( 68,182)( 69,181)( 70,180)( 71,179)
( 72,178)( 73,177)( 74,176)( 75,175)( 76,174)( 77,211)( 78,210)( 79,228)
( 80,227)( 81,226)( 82,225)( 83,224)( 84,223)( 85,222)( 86,221)( 87,220)
( 88,219)( 89,218)( 90,217)( 91,216)( 92,215)( 93,214)( 94,213)( 95,212)
( 96,192)( 97,191)( 98,209)( 99,208)(100,207)(101,206)(102,205)(103,204)
(104,203)(105,202)(106,201)(107,200)(108,199)(109,198)(110,197)(111,196)
(112,195)(113,194)(114,193);
s2 := Sym(230)!(  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);
s3 := Sym(230)!(229,230);
poly := sub<Sym(230)|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, 
s2*s3*s2*s3, s0*s1*s2*s1*s0*s1*s2*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
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*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*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*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*s0*s1*s0*s1*s0*s1*s0*s1 >;

to this polytope