Polytope of Type {4,4,3}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,4,3}*768a
Also Known As : 1T4(2,2), {{4,4}4,{4,3}}. if this polytope has another name.
Group : SmallGroup(768,1087581)
Rank : 4
Schlafli Type : {4,4,3}
Number of vertices, edges, etc : 16, 64, 48, 12
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 : {4,4,3}*384b
4-fold quotients : {4,4,3}*192a, {4,4,3}*192b
8-fold quotients : {2,4,3}*96
16-fold quotients : {4,2,3}*48, {2,4,3}*48
32-fold quotients : {2,2,3}*24
Covers (Minimal Covers in Boldface) :
None in this atlas.
Irregular Quotients (of which this is a minimal cover):
P/N, where N=<s1*s2*s1*s2> of order 2.
8 facets:
4 of 2-fold non-regular quotient of {4,4}*64
4 of {4,4}*64
12 vertex figures:
8 of 2-fold non-regular quotient of {4,3}*48
4 of {4,3}*48
P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1> of order 2.
8 facets:
4 of {4,4}*32
4 of {4,4}*64
8 vertex figures:
8 of {4,3}*48
P/N, where N=<s1*s2*s1*s2*s3*s2*s1*s2*s3> of order 2.
6 facets:
6 of {4,4}*64
12 vertex figures:
8 of {4,3}*24
4 of {4,3}*48
P/N, where N=<s0*s3*s2*s1*s0*s1*s2*s3> of order 2.
10 facets:
2 of {4,4}*64
8 of 2-fold non-regular quotient of {4,4}*64
8 vertex figures:
8 of {4,3}*48
P/N, where N=<s0*s1*s2*s1*s0*s2, s1*s2*s1*s2*s3*s2*s1*s2*s3> of order 4.
4 facets:
2 of 2-fold non-regular quotient of {4,4}*64
2 of {4,4}*64
8 vertex figures:
4 of {4,3}*24
4 of 2-fold non-regular quotient of {4,3}*48
P/N, where N=<s1*s2*s1*s2, s0*s1*s2*s1*s0*s2> of order 4.
6 facets:
4 of {4,2}*16
2 of {4,4}*64
8 vertex figures:
8 of 2-fold non-regular quotient of {4,3}*48
P/N, where N=<s1*s2*s1*s2, s1*s3*s2*s1*s2*s3> of order 4.
6 facets:
6 of 2-fold non-regular quotient of {4,4}*64
10 vertex figures:
8 of {2,3}*12
2 of {4,3}*48
P/N, where N=<s1*s2*s1*s2, s0*s1*s3*s2*s1*s0*s2*s3> of order 4.
5 facets:
4 of 2-fold non-regular quotient of {4,4}*64
1 of {4,4}*64
8 vertex figures:
8 of 2-fold non-regular quotient of {4,3}*48
P/N, where N=<s1*s2*s1*s2, s0*s3*s2*s1*s0*s1*s2*s3> of order 4.
6 facets:
2 of 2-fold non-regular quotient of {4,4}*64
4 of 2-fold non-regular quotient of {4,4}*64
6 vertex figures:
4 of 2-fold non-regular quotient of {4,3}*48
2 of {4,3}*48
P/N, where N=<s0*s1*s0*s1, s0*s2*s1*s0*s1*s2> of order 4.
8 facets:
4 of {2,4}*16
4 of 2-fold non-regular quotient of {4,4}*64
4 vertex figures:
4 of {4,3}*48
P/N, where N=<s0*s1*s0*s1, s1*s2*s1*s2, s0*s1*s2*s1*s0*s2> of order 8.
6 facets:
4 of {2,2}*8
2 of 2-fold non-regular quotient of {4,4}*64
4 vertex figures:
4 of 2-fold non-regular quotient of {4,3}*48
P/N, where N=<s1*s2*s1*s2, s0*s1*s2*s1*s0*s2, s1*s3*s2*s1*s2*s3> of order 8.
4 facets:
2 of {4,2}*16
2 of 2-fold non-regular quotient of {4,4}*64
6 vertex figures:
4 of {2,3}*12
2 of 2-fold non-regular quotient of {4,3}*48
P/N, where N=<s0*s1*s2*s1*s0*s2, s0*s1*s3*s2*s1*s0*s2*s3, s1*s2*s1*s2*s3*s2*s1*s2*s3> of order 8.
3 facets:
3 of 2-fold non-regular quotient of {4,4}*64
6 vertex figures:
2 of {4,3}*24
4 of {2,3}*12
Permutation Representation (GAP) :
s0 := ( 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);;
s1 := ( 1, 13)( 2, 14)( 3, 15)( 4, 16)( 5, 9)( 6, 10)( 7, 11)( 8, 12)( 17, 29)( 18, 30)( 19, 31)( 20, 32)( 21, 25)( 22, 26)( 23, 27)( 24, 28)( 33, 45)( 34, 46)( 35, 47)( 36, 48)( 37, 41)( 38, 42)( 39, 43)( 40, 44)( 49, 61)( 50, 62)( 51, 63)( 52, 64)( 53, 57)( 54, 58)( 55, 59)( 56, 60)( 65, 77)( 66, 78)( 67, 79)( 68, 80)( 69, 73)( 70, 74)( 71, 75)( 72, 76)( 81, 93)( 82, 94)( 83, 95)( 84, 96)( 85, 89)( 86, 90)( 87, 91)( 88, 92)( 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);;
s2 := ( 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);;
s3 := ( 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);;
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*s2*s3,
s0*s1*s0*s1*s0*s1*s0*s1, s1*s2*s1*s2*s1*s2*s1*s2,
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := 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);
s1 := Sym(192)!( 1, 13)( 2, 14)( 3, 15)( 4, 16)( 5, 9)( 6, 10)( 7, 11)( 8, 12)( 17, 29)( 18, 30)( 19, 31)( 20, 32)( 21, 25)( 22, 26)( 23, 27)( 24, 28)( 33, 45)( 34, 46)( 35, 47)( 36, 48)( 37, 41)( 38, 42)( 39, 43)( 40, 44)( 49, 61)( 50, 62)( 51, 63)( 52, 64)( 53, 57)( 54, 58)( 55, 59)( 56, 60)( 65, 77)( 66, 78)( 67, 79)( 68, 80)( 69, 73)( 70, 74)( 71, 75)( 72, 76)( 81, 93)( 82, 94)( 83, 95)( 84, 96)( 85, 89)( 86, 90)( 87, 91)( 88, 92)( 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);
s2 := 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);
s3 := 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);
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,
s2*s3*s2*s3*s2*s3, s0*s1*s0*s1*s0*s1*s0*s1,
s1*s2*s1*s2*s1*s2*s1*s2, s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1 >;
References : - Theorem 10B3, McMullen P., Schulte, E.; Abstract Regular Polytopes (Cambridge University Press, 2002)
to this polytope