Polytope of Type {3,4,4}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,4,4}*768a
Also Known As : Dual of 1T4(2,2), {{3,4},{4,4}4}. if this polytope has another name.
Group : SmallGroup(768,1087581)
Rank : 4
Schlafli Type : {3,4,4}
Number of vertices, edges, etc : 12, 48, 64, 16
Order of s0s1s2s3 : 12
Order of s0s1s2s3s2s1 : 4
Special Properties :
Universal
Locally Toroidal
Orientable
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 : {3,4,4}*384b
4-fold quotients : {3,4,4}*192a, {3,4,4}*192b
8-fold quotients : {3,4,2}*96
16-fold quotients : {3,2,4}*48, {3,4,2}*48
32-fold quotients : {3,2,2}*24
Covers (Minimal Covers in Boldface) :
None in this atlas.
Irregular Quotients (of which this is a minimal cover):
P/N, where N=<s0*s1*s0*s2*s1*s0*s2*s1> of order 2.
12 facets:
8 of 2-fold non-regular quotient of {3,4}*48
4 of {3,4}*48
8 vertex figures:
4 of {4,4}*64
4 of 2-fold non-regular quotient of {4,4}*64
P/N, where N=<s0*s1*s0*s2*s3*s2*s1*s0*s2*s3*s2*s1> of order 2.
8 facets:
8 of {3,4}*48
8 vertex figures:
4 of {4,4}*64
4 of {4,4}*32
P/N, where N=<s0*s1*s0*s2*s1*s0*s2*s1*s2> of order 2.
12 facets:
8 of {3,4}*24
4 of {3,4}*48
6 vertex figures:
6 of {4,4}*64
P/N, where N=<s2*s3*s2*s3> of order 2.
8 facets:
8 of {3,4}*48
10 vertex figures:
8 of 2-fold non-regular quotient of {4,4}*64
2 of {4,4}*64
P/N, where N=<s0*s1*s0*s2*s1*s0*s2*s1*s2, s0*s1*s0*s3*s2*s1*s0*s2*s1*s3> of order 4.
8 facets:
4 of {3,4}*24
4 of 2-fold non-regular quotient of {3,4}*48
4 vertex figures:
2 of {4,4}*64
2 of 2-fold non-regular quotient of {4,4}*64
P/N, where N=<s0*s1*s0*s2*s1*s0*s2*s1, s0*s1*s0*s3*s2*s1*s0*s2*s1*s3> of order 4.
8 facets:
8 of 2-fold non-regular quotient of {3,4}*48
6 vertex figures:
2 of {4,4}*64
4 of {2,4}*16
P/N, where N=<s1*s2*s1*s2, s0*s1*s2*s1*s0*s2> of order 4.
10 facets:
8 of {3,2}*12
2 of {3,4}*48
6 vertex figures:
6 of 2-fold non-regular quotient of {4,4}*64
P/N, where N=<s0*s1*s0*s2*s1*s0*s2*s1, s0*s1*s3*s2*s1*s0*s2*s3> of order 4.
8 facets:
8 of 2-fold non-regular quotient of {3,4}*48
5 vertex figures:
1 of {4,4}*64
4 of 2-fold non-regular quotient of {4,4}*64
P/N, where N=<s2*s3*s2*s3, s0*s1*s0*s2*s1*s0*s2*s1> of order 4.
6 facets:
4 of 2-fold non-regular quotient of {3,4}*48
2 of {3,4}*48
6 vertex figures:
4 of 2-fold non-regular quotient of {4,4}*64
2 of 2-fold non-regular quotient of {4,4}*64
P/N, where N=<s1*s2*s3*s2*s1*s3, s0*s1*s2*s3*s2*s1*s0*s3> of order 4.
4 facets:
4 of {3,4}*48
8 vertex figures:
4 of 2-fold non-regular quotient of {4,4}*64
4 of {4,2}*16
P/N, where N=<s1*s2*s3*s2*s1*s3, s0*s1*s0*s2*s1*s0*s2*s1, s0*s1*s2*s3*s2*s1*s0*s3> of order 8.
4 facets:
4 of 2-fold non-regular quotient of {3,4}*48
6 vertex figures:
2 of 2-fold non-regular quotient of {4,4}*64
4 of {2,2}*8
P/N, where N=<s1*s2*s1*s2, s0*s1*s2*s1*s0*s2, s0*s1*s0*s3*s2*s1*s0*s2*s1*s3> of order 8.
6 facets:
4 of {3,2}*12
2 of 2-fold non-regular quotient of {3,4}*48
4 vertex figures:
2 of 2-fold non-regular quotient of {4,4}*64
2 of {2,4}*16
P/N, where N=<s1*s3*s2*s1*s2*s3, s0*s1*s3*s2*s1*s0*s2*s3, s0*s1*s0*s2*s1*s0*s2*s1*s2> of order 8.
6 facets:
2 of {3,4}*24
4 of {3,2}*12
3 vertex figures:
3 of 2-fold non-regular quotient of {4,4}*64
Permutation Representation (GAP) :
s0 := ( 3, 4)( 7, 8)( 9, 13)( 10, 14)( 11, 16)( 12, 15)( 17, 33)( 18, 34)( 19, 36)( 20, 35)( 21, 37)( 22, 38)( 23, 40)( 24, 39)( 25, 45)( 26, 46)( 27, 48)( 28, 47)( 29, 41)( 30, 42)( 31, 44)( 32, 43)( 51, 52)( 55, 56)( 57, 61)( 58, 62)( 59, 64)( 60, 63)( 65, 81)( 66, 82)( 67, 84)( 68, 83)( 69, 85)( 70, 86)( 71, 88)( 72, 87)( 73, 93)( 74, 94)( 75, 96)( 76, 95)( 77, 89)( 78, 90)( 79, 92)( 80, 91)( 99,100)(103,104)(105,109)(106,110)(107,112)(108,111)(113,129)(114,130)(115,132)(116,131)(117,133)(118,134)(119,136)(120,135)(121,141)(122,142)(123,144)(124,143)(125,137)(126,138)(127,140)(128,139)(147,148)(151,152)(153,157)(154,158)(155,160)(156,159)(161,177)(162,178)(163,180)(164,179)(165,181)(166,182)(167,184)(168,183)(169,189)(170,190)(171,192)(172,191)(173,185)(174,186)(175,188)(176,187);;
s1 := ( 1, 33)( 2, 35)( 3, 34)( 4, 36)( 5, 41)( 6, 43)( 7, 42)( 8, 44)( 9, 37)( 10, 39)( 11, 38)( 12, 40)( 13, 45)( 14, 47)( 15, 46)( 16, 48)( 18, 19)( 21, 25)( 22, 27)( 23, 26)( 24, 28)( 30, 31)( 49, 81)( 50, 83)( 51, 82)( 52, 84)( 53, 89)( 54, 91)( 55, 90)( 56, 92)( 57, 85)( 58, 87)( 59, 86)( 60, 88)( 61, 93)( 62, 95)( 63, 94)( 64, 96)( 66, 67)( 69, 73)( 70, 75)( 71, 74)( 72, 76)( 78, 79)( 97,129)( 98,131)( 99,130)(100,132)(101,137)(102,139)(103,138)(104,140)(105,133)(106,135)(107,134)(108,136)(109,141)(110,143)(111,142)(112,144)(114,115)(117,121)(118,123)(119,122)(120,124)(126,127)(145,177)(146,179)(147,178)(148,180)(149,185)(150,187)(151,186)(152,188)(153,181)(154,183)(155,182)(156,184)(157,189)(158,191)(159,190)(160,192)(162,163)(165,169)(166,171)(167,170)(168,172)(174,175);;
s2 := ( 1, 5)( 2, 6)( 3, 7)( 4, 8)( 9, 13)( 10, 14)( 11, 15)( 12, 16)( 17, 21)( 18, 22)( 19, 23)( 20, 24)( 25, 29)( 26, 30)( 27, 31)( 28, 32)( 33, 37)( 34, 38)( 35, 39)( 36, 40)( 41, 45)( 42, 46)( 43, 47)( 44, 48)( 49, 53)( 50, 54)( 51, 55)( 52, 56)( 57, 61)( 58, 62)( 59, 63)( 60, 64)( 65, 69)( 66, 70)( 67, 71)( 68, 72)( 73, 77)( 74, 78)( 75, 79)( 76, 80)( 81, 85)( 82, 86)( 83, 87)( 84, 88)( 89, 93)( 90, 94)( 91, 95)( 92, 96)( 97,149)( 98,150)( 99,151)(100,152)(101,145)(102,146)(103,147)(104,148)(105,157)(106,158)(107,159)(108,160)(109,153)(110,154)(111,155)(112,156)(113,165)(114,166)(115,167)(116,168)(117,161)(118,162)(119,163)(120,164)(121,173)(122,174)(123,175)(124,176)(125,169)(126,170)(127,171)(128,172)(129,181)(130,182)(131,183)(132,184)(133,177)(134,178)(135,179)(136,180)(137,189)(138,190)(139,191)(140,192)(141,185)(142,186)(143,187)(144,188);;
s3 := ( 1, 97)( 2, 98)( 3, 99)( 4,100)( 5,102)( 6,101)( 7,104)( 8,103)( 9,107)( 10,108)( 11,105)( 12,106)( 13,112)( 14,111)( 15,110)( 16,109)( 17,113)( 18,114)( 19,115)( 20,116)( 21,118)( 22,117)( 23,120)( 24,119)( 25,123)( 26,124)( 27,121)( 28,122)( 29,128)( 30,127)( 31,126)( 32,125)( 33,129)( 34,130)( 35,131)( 36,132)( 37,134)( 38,133)( 39,136)( 40,135)( 41,139)( 42,140)( 43,137)( 44,138)( 45,144)( 46,143)( 47,142)( 48,141)( 49,145)( 50,146)( 51,147)( 52,148)( 53,150)( 54,149)( 55,152)( 56,151)( 57,155)( 58,156)( 59,153)( 60,154)( 61,160)( 62,159)( 63,158)( 64,157)( 65,161)( 66,162)( 67,163)( 68,164)( 69,166)( 70,165)( 71,168)( 72,167)( 73,171)( 74,172)( 75,169)( 76,170)( 77,176)( 78,175)( 79,174)( 80,173)( 81,177)( 82,178)( 83,179)( 84,180)( 85,182)( 86,181)( 87,184)( 88,183)( 89,187)( 90,188)( 91,185)( 92,186)( 93,192)( 94,191)( 95,190)( 96,189);;
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,
s1*s2*s1*s2*s1*s2*s1*s2, s2*s3*s2*s3*s2*s3*s2*s3,
s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(192)!( 3, 4)( 7, 8)( 9, 13)( 10, 14)( 11, 16)( 12, 15)( 17, 33)( 18, 34)( 19, 36)( 20, 35)( 21, 37)( 22, 38)( 23, 40)( 24, 39)( 25, 45)( 26, 46)( 27, 48)( 28, 47)( 29, 41)( 30, 42)( 31, 44)( 32, 43)( 51, 52)( 55, 56)( 57, 61)( 58, 62)( 59, 64)( 60, 63)( 65, 81)( 66, 82)( 67, 84)( 68, 83)( 69, 85)( 70, 86)( 71, 88)( 72, 87)( 73, 93)( 74, 94)( 75, 96)( 76, 95)( 77, 89)( 78, 90)( 79, 92)( 80, 91)( 99,100)(103,104)(105,109)(106,110)(107,112)(108,111)(113,129)(114,130)(115,132)(116,131)(117,133)(118,134)(119,136)(120,135)(121,141)(122,142)(123,144)(124,143)(125,137)(126,138)(127,140)(128,139)(147,148)(151,152)(153,157)(154,158)(155,160)(156,159)(161,177)(162,178)(163,180)(164,179)(165,181)(166,182)(167,184)(168,183)(169,189)(170,190)(171,192)(172,191)(173,185)(174,186)(175,188)(176,187);
s1 := Sym(192)!( 1, 33)( 2, 35)( 3, 34)( 4, 36)( 5, 41)( 6, 43)( 7, 42)( 8, 44)( 9, 37)( 10, 39)( 11, 38)( 12, 40)( 13, 45)( 14, 47)( 15, 46)( 16, 48)( 18, 19)( 21, 25)( 22, 27)( 23, 26)( 24, 28)( 30, 31)( 49, 81)( 50, 83)( 51, 82)( 52, 84)( 53, 89)( 54, 91)( 55, 90)( 56, 92)( 57, 85)( 58, 87)( 59, 86)( 60, 88)( 61, 93)( 62, 95)( 63, 94)( 64, 96)( 66, 67)( 69, 73)( 70, 75)( 71, 74)( 72, 76)( 78, 79)( 97,129)( 98,131)( 99,130)(100,132)(101,137)(102,139)(103,138)(104,140)(105,133)(106,135)(107,134)(108,136)(109,141)(110,143)(111,142)(112,144)(114,115)(117,121)(118,123)(119,122)(120,124)(126,127)(145,177)(146,179)(147,178)(148,180)(149,185)(150,187)(151,186)(152,188)(153,181)(154,183)(155,182)(156,184)(157,189)(158,191)(159,190)(160,192)(162,163)(165,169)(166,171)(167,170)(168,172)(174,175);
s2 := Sym(192)!( 1, 5)( 2, 6)( 3, 7)( 4, 8)( 9, 13)( 10, 14)( 11, 15)( 12, 16)( 17, 21)( 18, 22)( 19, 23)( 20, 24)( 25, 29)( 26, 30)( 27, 31)( 28, 32)( 33, 37)( 34, 38)( 35, 39)( 36, 40)( 41, 45)( 42, 46)( 43, 47)( 44, 48)( 49, 53)( 50, 54)( 51, 55)( 52, 56)( 57, 61)( 58, 62)( 59, 63)( 60, 64)( 65, 69)( 66, 70)( 67, 71)( 68, 72)( 73, 77)( 74, 78)( 75, 79)( 76, 80)( 81, 85)( 82, 86)( 83, 87)( 84, 88)( 89, 93)( 90, 94)( 91, 95)( 92, 96)( 97,149)( 98,150)( 99,151)(100,152)(101,145)(102,146)(103,147)(104,148)(105,157)(106,158)(107,159)(108,160)(109,153)(110,154)(111,155)(112,156)(113,165)(114,166)(115,167)(116,168)(117,161)(118,162)(119,163)(120,164)(121,173)(122,174)(123,175)(124,176)(125,169)(126,170)(127,171)(128,172)(129,181)(130,182)(131,183)(132,184)(133,177)(134,178)(135,179)(136,180)(137,189)(138,190)(139,191)(140,192)(141,185)(142,186)(143,187)(144,188);
s3 := Sym(192)!( 1, 97)( 2, 98)( 3, 99)( 4,100)( 5,102)( 6,101)( 7,104)( 8,103)( 9,107)( 10,108)( 11,105)( 12,106)( 13,112)( 14,111)( 15,110)( 16,109)( 17,113)( 18,114)( 19,115)( 20,116)( 21,118)( 22,117)( 23,120)( 24,119)( 25,123)( 26,124)( 27,121)( 28,122)( 29,128)( 30,127)( 31,126)( 32,125)( 33,129)( 34,130)( 35,131)( 36,132)( 37,134)( 38,133)( 39,136)( 40,135)( 41,139)( 42,140)( 43,137)( 44,138)( 45,144)( 46,143)( 47,142)( 48,141)( 49,145)( 50,146)( 51,147)( 52,148)( 53,150)( 54,149)( 55,152)( 56,151)( 57,155)( 58,156)( 59,153)( 60,154)( 61,160)( 62,159)( 63,158)( 64,157)( 65,161)( 66,162)( 67,163)( 68,164)( 69,166)( 70,165)( 71,168)( 72,167)( 73,171)( 74,172)( 75,169)( 76,170)( 77,176)( 78,175)( 79,174)( 80,173)( 81,177)( 82,178)( 83,179)( 84,180)( 85,182)( 86,181)( 87,184)( 88,183)( 89,187)( 90,188)( 91,185)( 92,186)( 93,192)( 94,191)( 95,190)( 96,189);
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, s1*s2*s1*s2*s1*s2*s1*s2,
s2*s3*s2*s3*s2*s3*s2*s3, s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2 >;
References : - Theorem 10B3, McMullen P., Schulte, E.; Abstract Regular Polytopes (Cambridge University Press, 2002)
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