Polytope of Type {2,76,6}

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

to this polytope