Polytope of Type {30,6,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {30,6,4}*1440a
if this polytope has a name.
Group : SmallGroup(1440,5360)
Rank : 4
Schlafli Type : {30,6,4}
Number of vertices, edges, etc : 30, 90, 12, 4
Order of s₀s₁s₂s₃ : 60
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 : {30,6,2}*720a
   3-fold quotients : {10,6,4}*480a
   5-fold quotients : {6,6,4}*288b
   6-fold quotients : {10,6,2}*240
   9-fold quotients : {10,2,4}*160
   10-fold quotients : {6,6,2}*144b
   15-fold quotients : {2,6,4}*96a
   18-fold quotients : {5,2,4}*80, {10,2,2}*80
   20-fold quotients : {6,3,2}*72
   30-fold quotients : {2,6,2}*48
   36-fold quotients : {5,2,2}*40
   45-fold quotients : {2,2,4}*32
   60-fold quotients : {2,3,2}*24
   90-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := (  2,  5)(  3,  4)(  7, 10)(  8,  9)( 12, 15)( 13, 14)( 16, 31)( 17, 35)
( 18, 34)( 19, 33)( 20, 32)( 21, 36)( 22, 40)( 23, 39)( 24, 38)( 25, 37)
( 26, 41)( 27, 45)( 28, 44)( 29, 43)( 30, 42)( 47, 50)( 48, 49)( 52, 55)
( 53, 54)( 57, 60)( 58, 59)( 61, 76)( 62, 80)( 63, 79)( 64, 78)( 65, 77)
( 66, 81)( 67, 85)( 68, 84)( 69, 83)( 70, 82)( 71, 86)( 72, 90)( 73, 89)
( 74, 88)( 75, 87)( 92, 95)( 93, 94)( 97,100)( 98, 99)(102,105)(103,104)
(106,121)(107,125)(108,124)(109,123)(110,122)(111,126)(112,130)(113,129)
(114,128)(115,127)(116,131)(117,135)(118,134)(119,133)(120,132)(137,140)
(138,139)(142,145)(143,144)(147,150)(148,149)(151,166)(152,170)(153,169)
(154,168)(155,167)(156,171)(157,175)(158,174)(159,173)(160,172)(161,176)
(162,180)(163,179)(164,178)(165,177);;
s1 := (  1, 17)(  2, 16)(  3, 20)(  4, 19)(  5, 18)(  6, 27)(  7, 26)(  8, 30)
(  9, 29)( 10, 28)( 11, 22)( 12, 21)( 13, 25)( 14, 24)( 15, 23)( 31, 32)
( 33, 35)( 36, 42)( 37, 41)( 38, 45)( 39, 44)( 40, 43)( 46, 62)( 47, 61)
( 48, 65)( 49, 64)( 50, 63)( 51, 72)( 52, 71)( 53, 75)( 54, 74)( 55, 73)
( 56, 67)( 57, 66)( 58, 70)( 59, 69)( 60, 68)( 76, 77)( 78, 80)( 81, 87)
( 82, 86)( 83, 90)( 84, 89)( 85, 88)( 91,107)( 92,106)( 93,110)( 94,109)
( 95,108)( 96,117)( 97,116)( 98,120)( 99,119)(100,118)(101,112)(102,111)
(103,115)(104,114)(105,113)(121,122)(123,125)(126,132)(127,131)(128,135)
(129,134)(130,133)(136,152)(137,151)(138,155)(139,154)(140,153)(141,162)
(142,161)(143,165)(144,164)(145,163)(146,157)(147,156)(148,160)(149,159)
(150,158)(166,167)(168,170)(171,177)(172,176)(173,180)(174,179)(175,178);;
s2 := (  1,  6)(  2,  7)(  3,  8)(  4,  9)(  5, 10)( 16, 36)( 17, 37)( 18, 38)
( 19, 39)( 20, 40)( 21, 31)( 22, 32)( 23, 33)( 24, 34)( 25, 35)( 26, 41)
( 27, 42)( 28, 43)( 29, 44)( 30, 45)( 46, 51)( 47, 52)( 48, 53)( 49, 54)
( 50, 55)( 61, 81)( 62, 82)( 63, 83)( 64, 84)( 65, 85)( 66, 76)( 67, 77)
( 68, 78)( 69, 79)( 70, 80)( 71, 86)( 72, 87)( 73, 88)( 74, 89)( 75, 90)
( 91,141)( 92,142)( 93,143)( 94,144)( 95,145)( 96,136)( 97,137)( 98,138)
( 99,139)(100,140)(101,146)(102,147)(103,148)(104,149)(105,150)(106,171)
(107,172)(108,173)(109,174)(110,175)(111,166)(112,167)(113,168)(114,169)
(115,170)(116,176)(117,177)(118,178)(119,179)(120,180)(121,156)(122,157)
(123,158)(124,159)(125,160)(126,151)(127,152)(128,153)(129,154)(130,155)
(131,161)(132,162)(133,163)(134,164)(135,165);;
s3 := (  1, 91)(  2, 92)(  3, 93)(  4, 94)(  5, 95)(  6, 96)(  7, 97)(  8, 98)
(  9, 99)( 10,100)( 11,101)( 12,102)( 13,103)( 14,104)( 15,105)( 16,106)
( 17,107)( 18,108)( 19,109)( 20,110)( 21,111)( 22,112)( 23,113)( 24,114)
( 25,115)( 26,116)( 27,117)( 28,118)( 29,119)( 30,120)( 31,121)( 32,122)
( 33,123)( 34,124)( 35,125)( 36,126)( 37,127)( 38,128)( 39,129)( 40,130)
( 41,131)( 42,132)( 43,133)( 44,134)( 45,135)( 46,136)( 47,137)( 48,138)
( 49,139)( 50,140)( 51,141)( 52,142)( 53,143)( 54,144)( 55,145)( 56,146)
( 57,147)( 58,148)( 59,149)( 60,150)( 61,151)( 62,152)( 63,153)( 64,154)
( 65,155)( 66,156)( 67,157)( 68,158)( 69,159)( 70,160)( 71,161)( 72,162)
( 73,163)( 74,164)( 75,165)( 76,166)( 77,167)( 78,168)( 79,169)( 80,170)
( 81,171)( 82,172)( 83,173)( 84,174)( 85,175)( 86,176)( 87,177)( 88,178)
( 89,179)( 90,180);;
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, s1*s2*s3*s2*s1*s2*s3*s2, 
s2*s3*s2*s3*s2*s3*s2*s3, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s0*s1*s2*s1*s0*s1*s2*s0*s1*s2*s1*s0*s1, 
s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(180)!(  2,  5)(  3,  4)(  7, 10)(  8,  9)( 12, 15)( 13, 14)( 16, 31)
( 17, 35)( 18, 34)( 19, 33)( 20, 32)( 21, 36)( 22, 40)( 23, 39)( 24, 38)
( 25, 37)( 26, 41)( 27, 45)( 28, 44)( 29, 43)( 30, 42)( 47, 50)( 48, 49)
( 52, 55)( 53, 54)( 57, 60)( 58, 59)( 61, 76)( 62, 80)( 63, 79)( 64, 78)
( 65, 77)( 66, 81)( 67, 85)( 68, 84)( 69, 83)( 70, 82)( 71, 86)( 72, 90)
( 73, 89)( 74, 88)( 75, 87)( 92, 95)( 93, 94)( 97,100)( 98, 99)(102,105)
(103,104)(106,121)(107,125)(108,124)(109,123)(110,122)(111,126)(112,130)
(113,129)(114,128)(115,127)(116,131)(117,135)(118,134)(119,133)(120,132)
(137,140)(138,139)(142,145)(143,144)(147,150)(148,149)(151,166)(152,170)
(153,169)(154,168)(155,167)(156,171)(157,175)(158,174)(159,173)(160,172)
(161,176)(162,180)(163,179)(164,178)(165,177);
s1 := Sym(180)!(  1, 17)(  2, 16)(  3, 20)(  4, 19)(  5, 18)(  6, 27)(  7, 26)
(  8, 30)(  9, 29)( 10, 28)( 11, 22)( 12, 21)( 13, 25)( 14, 24)( 15, 23)
( 31, 32)( 33, 35)( 36, 42)( 37, 41)( 38, 45)( 39, 44)( 40, 43)( 46, 62)
( 47, 61)( 48, 65)( 49, 64)( 50, 63)( 51, 72)( 52, 71)( 53, 75)( 54, 74)
( 55, 73)( 56, 67)( 57, 66)( 58, 70)( 59, 69)( 60, 68)( 76, 77)( 78, 80)
( 81, 87)( 82, 86)( 83, 90)( 84, 89)( 85, 88)( 91,107)( 92,106)( 93,110)
( 94,109)( 95,108)( 96,117)( 97,116)( 98,120)( 99,119)(100,118)(101,112)
(102,111)(103,115)(104,114)(105,113)(121,122)(123,125)(126,132)(127,131)
(128,135)(129,134)(130,133)(136,152)(137,151)(138,155)(139,154)(140,153)
(141,162)(142,161)(143,165)(144,164)(145,163)(146,157)(147,156)(148,160)
(149,159)(150,158)(166,167)(168,170)(171,177)(172,176)(173,180)(174,179)
(175,178);
s2 := Sym(180)!(  1,  6)(  2,  7)(  3,  8)(  4,  9)(  5, 10)( 16, 36)( 17, 37)
( 18, 38)( 19, 39)( 20, 40)( 21, 31)( 22, 32)( 23, 33)( 24, 34)( 25, 35)
( 26, 41)( 27, 42)( 28, 43)( 29, 44)( 30, 45)( 46, 51)( 47, 52)( 48, 53)
( 49, 54)( 50, 55)( 61, 81)( 62, 82)( 63, 83)( 64, 84)( 65, 85)( 66, 76)
( 67, 77)( 68, 78)( 69, 79)( 70, 80)( 71, 86)( 72, 87)( 73, 88)( 74, 89)
( 75, 90)( 91,141)( 92,142)( 93,143)( 94,144)( 95,145)( 96,136)( 97,137)
( 98,138)( 99,139)(100,140)(101,146)(102,147)(103,148)(104,149)(105,150)
(106,171)(107,172)(108,173)(109,174)(110,175)(111,166)(112,167)(113,168)
(114,169)(115,170)(116,176)(117,177)(118,178)(119,179)(120,180)(121,156)
(122,157)(123,158)(124,159)(125,160)(126,151)(127,152)(128,153)(129,154)
(130,155)(131,161)(132,162)(133,163)(134,164)(135,165);
s3 := Sym(180)!(  1, 91)(  2, 92)(  3, 93)(  4, 94)(  5, 95)(  6, 96)(  7, 97)
(  8, 98)(  9, 99)( 10,100)( 11,101)( 12,102)( 13,103)( 14,104)( 15,105)
( 16,106)( 17,107)( 18,108)( 19,109)( 20,110)( 21,111)( 22,112)( 23,113)
( 24,114)( 25,115)( 26,116)( 27,117)( 28,118)( 29,119)( 30,120)( 31,121)
( 32,122)( 33,123)( 34,124)( 35,125)( 36,126)( 37,127)( 38,128)( 39,129)
( 40,130)( 41,131)( 42,132)( 43,133)( 44,134)( 45,135)( 46,136)( 47,137)
( 48,138)( 49,139)( 50,140)( 51,141)( 52,142)( 53,143)( 54,144)( 55,145)
( 56,146)( 57,147)( 58,148)( 59,149)( 60,150)( 61,151)( 62,152)( 63,153)
( 64,154)( 65,155)( 66,156)( 67,157)( 68,158)( 69,159)( 70,160)( 71,161)
( 72,162)( 73,163)( 74,164)( 75,165)( 76,166)( 77,167)( 78,168)( 79,169)
( 80,170)( 81,171)( 82,172)( 83,173)( 84,174)( 85,175)( 86,176)( 87,177)
( 88,178)( 89,179)( 90,180);
poly := sub<Sym(180)|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, 
s1*s2*s3*s2*s1*s2*s3*s2, s2*s3*s2*s3*s2*s3*s2*s3, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s0*s1*s2*s1*s0*s1*s2*s0*s1*s2*s1*s0*s1, 
s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;

References : None.
to this polytope