Polytope of Type {4,6,4}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,6,4}*192g
Also Known As : {{4,6}3,{6,4}3}. if this polytope has another name.
Group : SmallGroup(192,1538)
Rank : 4
Schlafli Type : {4,6,4}
Number of vertices, edges, etc : 4, 12, 12, 4
Order of s0s1s2s3 : 6
Order of s0s1s2s3s2s1 : 2
Special Properties :
Universal
Non-Orientable
Flat
Self-Dual
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{4,6,4,2} of size 384
Vertex Figure Of :
{2,4,6,4} of size 384
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {4,3,4}*96
Covers (Minimal Covers in Boldface) :
2-fold covers : {4,6,4}*384e, {4,6,4}*384f
3-fold covers : {4,18,4}*576g
4-fold covers : {4,6,8}*768f, {8,6,4}*768f, {4,6,8}*768h, {8,6,4}*768h, {4,6,4}*768i, {4,6,4}*768k, {4,6,4}*768l
5-fold covers : {4,30,4}*960g
6-fold covers : {4,18,4}*1152e, {4,18,4}*1152f, {4,6,12}*1152g, {4,6,12}*1152h, {12,6,4}*1152e, {12,6,4}*1152h
7-fold covers : {4,42,4}*1344g
9-fold covers : {4,54,4}*1728g
10-fold covers : {4,6,20}*1920d, {20,6,4}*1920c, {4,30,4}*1920e, {4,30,4}*1920f
Permutation Representation (GAP) :
s0 := ( 1, 50)( 2, 49)( 3, 52)( 4, 51)( 5, 54)( 6, 53)( 7, 56)( 8, 55)
( 9, 58)( 10, 57)( 11, 60)( 12, 59)( 13, 62)( 14, 61)( 15, 64)( 16, 63)
( 17, 66)( 18, 65)( 19, 68)( 20, 67)( 21, 70)( 22, 69)( 23, 72)( 24, 71)
( 25, 74)( 26, 73)( 27, 76)( 28, 75)( 29, 78)( 30, 77)( 31, 80)( 32, 79)
( 33, 82)( 34, 81)( 35, 84)( 36, 83)( 37, 86)( 38, 85)( 39, 88)( 40, 87)
( 41, 90)( 42, 89)( 43, 92)( 44, 91)( 45, 94)( 46, 93)( 47, 96)( 48, 95)
( 97,146)( 98,145)( 99,148)(100,147)(101,150)(102,149)(103,152)(104,151)
(105,154)(106,153)(107,156)(108,155)(109,158)(110,157)(111,160)(112,159)
(113,162)(114,161)(115,164)(116,163)(117,166)(118,165)(119,168)(120,167)
(121,170)(122,169)(123,172)(124,171)(125,174)(126,173)(127,176)(128,175)
(129,178)(130,177)(131,180)(132,179)(133,182)(134,181)(135,184)(136,183)
(137,186)(138,185)(139,188)(140,187)(141,190)(142,189)(143,192)(144,191);;
s1 := ( 1,113)( 2,116)( 3,115)( 4,114)( 5,121)( 6,124)( 7,123)( 8,122)
( 9,117)( 10,120)( 11,119)( 12,118)( 13,125)( 14,128)( 15,127)( 16,126)
( 17, 97)( 18,100)( 19, 99)( 20, 98)( 21,105)( 22,108)( 23,107)( 24,106)
( 25,101)( 26,104)( 27,103)( 28,102)( 29,109)( 30,112)( 31,111)( 32,110)
( 33,129)( 34,132)( 35,131)( 36,130)( 37,137)( 38,140)( 39,139)( 40,138)
( 41,133)( 42,136)( 43,135)( 44,134)( 45,141)( 46,144)( 47,143)( 48,142)
( 49,161)( 50,164)( 51,163)( 52,162)( 53,169)( 54,172)( 55,171)( 56,170)
( 57,165)( 58,168)( 59,167)( 60,166)( 61,173)( 62,176)( 63,175)( 64,174)
( 65,145)( 66,148)( 67,147)( 68,146)( 69,153)( 70,156)( 71,155)( 72,154)
( 73,149)( 74,152)( 75,151)( 76,150)( 77,157)( 78,160)( 79,159)( 80,158)
( 81,177)( 82,180)( 83,179)( 84,178)( 85,185)( 86,188)( 87,187)( 88,186)
( 89,181)( 90,184)( 91,183)( 92,182)( 93,189)( 94,192)( 95,191)( 96,190);;
s2 := ( 1,145)( 2,146)( 3,148)( 4,147)( 5,157)( 6,158)( 7,160)( 8,159)
( 9,153)( 10,154)( 11,156)( 12,155)( 13,149)( 14,150)( 15,152)( 16,151)
( 17,177)( 18,178)( 19,180)( 20,179)( 21,189)( 22,190)( 23,192)( 24,191)
( 25,185)( 26,186)( 27,188)( 28,187)( 29,181)( 30,182)( 31,184)( 32,183)
( 33,161)( 34,162)( 35,164)( 36,163)( 37,173)( 38,174)( 39,176)( 40,175)
( 41,169)( 42,170)( 43,172)( 44,171)( 45,165)( 46,166)( 47,168)( 48,167)
( 49, 97)( 50, 98)( 51,100)( 52, 99)( 53,109)( 54,110)( 55,112)( 56,111)
( 57,105)( 58,106)( 59,108)( 60,107)( 61,101)( 62,102)( 63,104)( 64,103)
( 65,129)( 66,130)( 67,132)( 68,131)( 69,141)( 70,142)( 71,144)( 72,143)
( 73,137)( 74,138)( 75,140)( 76,139)( 77,133)( 78,134)( 79,136)( 80,135)
( 81,113)( 82,114)( 83,116)( 84,115)( 85,125)( 86,126)( 87,128)( 88,127)
( 89,121)( 90,122)( 91,124)( 92,123)( 93,117)( 94,118)( 95,120)( 96,119);;
s3 := ( 1, 61)( 2, 62)( 3, 63)( 4, 64)( 5, 57)( 6, 58)( 7, 59)( 8, 60)
( 9, 53)( 10, 54)( 11, 55)( 12, 56)( 13, 49)( 14, 50)( 15, 51)( 16, 52)
( 17, 77)( 18, 78)( 19, 79)( 20, 80)( 21, 73)( 22, 74)( 23, 75)( 24, 76)
( 25, 69)( 26, 70)( 27, 71)( 28, 72)( 29, 65)( 30, 66)( 31, 67)( 32, 68)
( 33, 93)( 34, 94)( 35, 95)( 36, 96)( 37, 89)( 38, 90)( 39, 91)( 40, 92)
( 41, 85)( 42, 86)( 43, 87)( 44, 88)( 45, 81)( 46, 82)( 47, 83)( 48, 84)
( 97,157)( 98,158)( 99,159)(100,160)(101,153)(102,154)(103,155)(104,156)
(105,149)(106,150)(107,151)(108,152)(109,145)(110,146)(111,147)(112,148)
(113,173)(114,174)(115,175)(116,176)(117,169)(118,170)(119,171)(120,172)
(121,165)(122,166)(123,167)(124,168)(125,161)(126,162)(127,163)(128,164)
(129,189)(130,190)(131,191)(132,192)(133,185)(134,186)(135,187)(136,188)
(137,181)(138,182)(139,183)(140,184)(141,177)(142,178)(143,179)(144,180);;
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, s0*s1*s0*s1*s0*s1*s0*s1,
s2*s3*s2*s3*s2*s3*s2*s3, s0*s1*s2*s0*s1*s2*s0*s1*s2,
s1*s2*s3*s1*s2*s3*s1*s2*s3 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(192)!( 1, 50)( 2, 49)( 3, 52)( 4, 51)( 5, 54)( 6, 53)( 7, 56)
( 8, 55)( 9, 58)( 10, 57)( 11, 60)( 12, 59)( 13, 62)( 14, 61)( 15, 64)
( 16, 63)( 17, 66)( 18, 65)( 19, 68)( 20, 67)( 21, 70)( 22, 69)( 23, 72)
( 24, 71)( 25, 74)( 26, 73)( 27, 76)( 28, 75)( 29, 78)( 30, 77)( 31, 80)
( 32, 79)( 33, 82)( 34, 81)( 35, 84)( 36, 83)( 37, 86)( 38, 85)( 39, 88)
( 40, 87)( 41, 90)( 42, 89)( 43, 92)( 44, 91)( 45, 94)( 46, 93)( 47, 96)
( 48, 95)( 97,146)( 98,145)( 99,148)(100,147)(101,150)(102,149)(103,152)
(104,151)(105,154)(106,153)(107,156)(108,155)(109,158)(110,157)(111,160)
(112,159)(113,162)(114,161)(115,164)(116,163)(117,166)(118,165)(119,168)
(120,167)(121,170)(122,169)(123,172)(124,171)(125,174)(126,173)(127,176)
(128,175)(129,178)(130,177)(131,180)(132,179)(133,182)(134,181)(135,184)
(136,183)(137,186)(138,185)(139,188)(140,187)(141,190)(142,189)(143,192)
(144,191);
s1 := Sym(192)!( 1,113)( 2,116)( 3,115)( 4,114)( 5,121)( 6,124)( 7,123)
( 8,122)( 9,117)( 10,120)( 11,119)( 12,118)( 13,125)( 14,128)( 15,127)
( 16,126)( 17, 97)( 18,100)( 19, 99)( 20, 98)( 21,105)( 22,108)( 23,107)
( 24,106)( 25,101)( 26,104)( 27,103)( 28,102)( 29,109)( 30,112)( 31,111)
( 32,110)( 33,129)( 34,132)( 35,131)( 36,130)( 37,137)( 38,140)( 39,139)
( 40,138)( 41,133)( 42,136)( 43,135)( 44,134)( 45,141)( 46,144)( 47,143)
( 48,142)( 49,161)( 50,164)( 51,163)( 52,162)( 53,169)( 54,172)( 55,171)
( 56,170)( 57,165)( 58,168)( 59,167)( 60,166)( 61,173)( 62,176)( 63,175)
( 64,174)( 65,145)( 66,148)( 67,147)( 68,146)( 69,153)( 70,156)( 71,155)
( 72,154)( 73,149)( 74,152)( 75,151)( 76,150)( 77,157)( 78,160)( 79,159)
( 80,158)( 81,177)( 82,180)( 83,179)( 84,178)( 85,185)( 86,188)( 87,187)
( 88,186)( 89,181)( 90,184)( 91,183)( 92,182)( 93,189)( 94,192)( 95,191)
( 96,190);
s2 := Sym(192)!( 1,145)( 2,146)( 3,148)( 4,147)( 5,157)( 6,158)( 7,160)
( 8,159)( 9,153)( 10,154)( 11,156)( 12,155)( 13,149)( 14,150)( 15,152)
( 16,151)( 17,177)( 18,178)( 19,180)( 20,179)( 21,189)( 22,190)( 23,192)
( 24,191)( 25,185)( 26,186)( 27,188)( 28,187)( 29,181)( 30,182)( 31,184)
( 32,183)( 33,161)( 34,162)( 35,164)( 36,163)( 37,173)( 38,174)( 39,176)
( 40,175)( 41,169)( 42,170)( 43,172)( 44,171)( 45,165)( 46,166)( 47,168)
( 48,167)( 49, 97)( 50, 98)( 51,100)( 52, 99)( 53,109)( 54,110)( 55,112)
( 56,111)( 57,105)( 58,106)( 59,108)( 60,107)( 61,101)( 62,102)( 63,104)
( 64,103)( 65,129)( 66,130)( 67,132)( 68,131)( 69,141)( 70,142)( 71,144)
( 72,143)( 73,137)( 74,138)( 75,140)( 76,139)( 77,133)( 78,134)( 79,136)
( 80,135)( 81,113)( 82,114)( 83,116)( 84,115)( 85,125)( 86,126)( 87,128)
( 88,127)( 89,121)( 90,122)( 91,124)( 92,123)( 93,117)( 94,118)( 95,120)
( 96,119);
s3 := Sym(192)!( 1, 61)( 2, 62)( 3, 63)( 4, 64)( 5, 57)( 6, 58)( 7, 59)
( 8, 60)( 9, 53)( 10, 54)( 11, 55)( 12, 56)( 13, 49)( 14, 50)( 15, 51)
( 16, 52)( 17, 77)( 18, 78)( 19, 79)( 20, 80)( 21, 73)( 22, 74)( 23, 75)
( 24, 76)( 25, 69)( 26, 70)( 27, 71)( 28, 72)( 29, 65)( 30, 66)( 31, 67)
( 32, 68)( 33, 93)( 34, 94)( 35, 95)( 36, 96)( 37, 89)( 38, 90)( 39, 91)
( 40, 92)( 41, 85)( 42, 86)( 43, 87)( 44, 88)( 45, 81)( 46, 82)( 47, 83)
( 48, 84)( 97,157)( 98,158)( 99,159)(100,160)(101,153)(102,154)(103,155)
(104,156)(105,149)(106,150)(107,151)(108,152)(109,145)(110,146)(111,147)
(112,148)(113,173)(114,174)(115,175)(116,176)(117,169)(118,170)(119,171)
(120,172)(121,165)(122,166)(123,167)(124,168)(125,161)(126,162)(127,163)
(128,164)(129,189)(130,190)(131,191)(132,192)(133,185)(134,186)(135,187)
(136,188)(137,181)(138,182)(139,183)(140,184)(141,177)(142,178)(143,179)
(144,180);
poly := sub<Sym(192)|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, s2*s3*s2*s3*s2*s3*s2*s3,
s0*s1*s2*s0*s1*s2*s0*s1*s2, s1*s2*s3*s1*s2*s3*s1*s2*s3 >;
References : None.
to this polytope