Polytope of Type {4,4,6}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,4,6}*384d
if this polytope has a name.
Group : SmallGroup(384,20051)
Rank : 4
Schlafli Type : {4,4,6}
Number of vertices, edges, etc : 4, 16, 24, 12
Order of s0s1s2s3 : 12
Order of s0s1s2s3s2s1 : 2
Special Properties :
Universal
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{4,4,6,2} of size 768
Vertex Figure Of :
{2,4,4,6} of size 768
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {4,4,3}*192b, {2,4,6}*192
4-fold quotients : {4,2,6}*96, {2,4,3}*96, {2,4,6}*96b, {2,4,6}*96c
8-fold quotients : {4,2,3}*48, {2,4,3}*48, {2,2,6}*48
12-fold quotients : {4,2,2}*32
16-fold quotients : {2,2,3}*24
24-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
2-fold covers : {4,4,6}*768e, {4,4,12}*768e, {4,4,12}*768f, {4,8,6}*768c, {8,4,6}*768c, {4,8,6}*768d
3-fold covers : {4,4,18}*1152d, {12,4,6}*1152c, {4,12,6}*1152g, {4,12,6}*1152j
5-fold covers : {20,4,6}*1920b, {4,20,6}*1920c, {4,4,30}*1920d
Irregular Quotients (of which this is a minimal cover):
P/N, where N=<s1*s3*s2*s1*s2*s3> of order 2.
8 facets:
4 of {4,4}*32
4 of {4,2}*16
4 vertex figures:
4 of 2-fold non-regular quotient of {4,6}*96
Permutation Representation (GAP) :
s0 := ( 1, 49)( 2, 50)( 3, 51)( 4, 52)( 5, 53)( 6, 54)( 7, 55)( 8, 56)( 9, 57)( 10, 58)( 11, 59)( 12, 60)( 13, 61)( 14, 62)( 15, 63)( 16, 64)( 17, 65)( 18, 66)( 19, 67)( 20, 68)( 21, 69)( 22, 70)( 23, 71)( 24, 72)( 25, 85)( 26, 86)( 27, 87)( 28, 88)( 29, 89)( 30, 90)( 31, 91)( 32, 92)( 33, 93)( 34, 94)( 35, 95)( 36, 96)( 37, 73)( 38, 74)( 39, 75)( 40, 76)( 41, 77)( 42, 78)( 43, 79)( 44, 80)( 45, 81)( 46, 82)( 47, 83)( 48, 84)( 97,145)( 98,146)( 99,147)(100,148)(101,149)(102,150)(103,151)(104,152)(105,153)(106,154)(107,155)(108,156)(109,157)(110,158)(111,159)(112,160)(113,161)(114,162)(115,163)(116,164)(117,165)(118,166)(119,167)(120,168)(121,181)(122,182)(123,183)(124,184)(125,185)(126,186)(127,187)(128,188)(129,189)(130,190)(131,191)(132,192)(133,169)(134,170)(135,171)(136,172)(137,173)(138,174)(139,175)(140,176)(141,177)(142,178)(143,179)(144,180);;
s1 := ( 1, 27)( 2, 28)( 3, 25)( 4, 26)( 5, 31)( 6, 32)( 7, 29)( 8, 30)( 9, 35)( 10, 36)( 11, 33)( 12, 34)( 13, 39)( 14, 40)( 15, 37)( 16, 38)( 17, 43)( 18, 44)( 19, 41)( 20, 42)( 21, 47)( 22, 48)( 23, 45)( 24, 46)( 49, 75)( 50, 76)( 51, 73)( 52, 74)( 53, 79)( 54, 80)( 55, 77)( 56, 78)( 57, 83)( 58, 84)( 59, 81)( 60, 82)( 61, 87)( 62, 88)( 63, 85)( 64, 86)( 65, 91)( 66, 92)( 67, 89)( 68, 90)( 69, 95)( 70, 96)( 71, 93)( 72, 94)( 97,123)( 98,124)( 99,121)(100,122)(101,127)(102,128)(103,125)(104,126)(105,131)(106,132)(107,129)(108,130)(109,135)(110,136)(111,133)(112,134)(113,139)(114,140)(115,137)(116,138)(117,143)(118,144)(119,141)(120,142)(145,171)(146,172)(147,169)(148,170)(149,175)(150,176)(151,173)(152,174)(153,179)(154,180)(155,177)(156,178)(157,183)(158,184)(159,181)(160,182)(161,187)(162,188)(163,185)(164,186)(165,191)(166,192)(167,189)(168,190);;
s2 := ( 2, 3)( 5, 9)( 6, 11)( 7, 10)( 8, 12)( 14, 15)( 17, 21)( 18, 23)( 19, 22)( 20, 24)( 26, 27)( 29, 33)( 30, 35)( 31, 34)( 32, 36)( 38, 39)( 41, 45)( 42, 47)( 43, 46)( 44, 48)( 50, 51)( 53, 57)( 54, 59)( 55, 58)( 56, 60)( 62, 63)( 65, 69)( 66, 71)( 67, 70)( 68, 72)( 74, 75)( 77, 81)( 78, 83)( 79, 82)( 80, 84)( 86, 87)( 89, 93)( 90, 95)( 91, 94)( 92, 96)( 98, 99)(101,105)(102,107)(103,106)(104,108)(110,111)(113,117)(114,119)(115,118)(116,120)(122,123)(125,129)(126,131)(127,130)(128,132)(134,135)(137,141)(138,143)(139,142)(140,144)(146,147)(149,153)(150,155)(151,154)(152,156)(158,159)(161,165)(162,167)(163,166)(164,168)(170,171)(173,177)(174,179)(175,178)(176,180)(182,183)(185,189)(186,191)(187,190)(188,192);;
s3 := ( 1,105)( 2,108)( 3,107)( 4,106)( 5,101)( 6,104)( 7,103)( 8,102)( 9, 97)( 10,100)( 11, 99)( 12, 98)( 13,117)( 14,120)( 15,119)( 16,118)( 17,113)( 18,116)( 19,115)( 20,114)( 21,109)( 22,112)( 23,111)( 24,110)( 25,129)( 26,132)( 27,131)( 28,130)( 29,125)( 30,128)( 31,127)( 32,126)( 33,121)( 34,124)( 35,123)( 36,122)( 37,141)( 38,144)( 39,143)( 40,142)( 41,137)( 42,140)( 43,139)( 44,138)( 45,133)( 46,136)( 47,135)( 48,134)( 49,153)( 50,156)( 51,155)( 52,154)( 53,149)( 54,152)( 55,151)( 56,150)( 57,145)( 58,148)( 59,147)( 60,146)( 61,165)( 62,168)( 63,167)( 64,166)( 65,161)( 66,164)( 67,163)( 68,162)( 69,157)( 70,160)( 71,159)( 72,158)( 73,177)( 74,180)( 75,179)( 76,178)( 77,173)( 78,176)( 79,175)( 80,174)( 81,169)( 82,172)( 83,171)( 84,170)( 85,189)( 86,192)( 87,191)( 88,190)( 89,185)( 90,188)( 91,187)( 92,186)( 93,181)( 94,184)( 95,183)( 96,182);;
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,
s0*s1*s2*s1*s0*s1*s2*s1, s1*s2*s1*s2*s1*s2*s1*s2,
s1*s2*s3*s2*s3*s2*s1*s2*s3*s2*s3*s2,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(192)!( 1, 49)( 2, 50)( 3, 51)( 4, 52)( 5, 53)( 6, 54)( 7, 55)( 8, 56)( 9, 57)( 10, 58)( 11, 59)( 12, 60)( 13, 61)( 14, 62)( 15, 63)( 16, 64)( 17, 65)( 18, 66)( 19, 67)( 20, 68)( 21, 69)( 22, 70)( 23, 71)( 24, 72)( 25, 85)( 26, 86)( 27, 87)( 28, 88)( 29, 89)( 30, 90)( 31, 91)( 32, 92)( 33, 93)( 34, 94)( 35, 95)( 36, 96)( 37, 73)( 38, 74)( 39, 75)( 40, 76)( 41, 77)( 42, 78)( 43, 79)( 44, 80)( 45, 81)( 46, 82)( 47, 83)( 48, 84)( 97,145)( 98,146)( 99,147)(100,148)(101,149)(102,150)(103,151)(104,152)(105,153)(106,154)(107,155)(108,156)(109,157)(110,158)(111,159)(112,160)(113,161)(114,162)(115,163)(116,164)(117,165)(118,166)(119,167)(120,168)(121,181)(122,182)(123,183)(124,184)(125,185)(126,186)(127,187)(128,188)(129,189)(130,190)(131,191)(132,192)(133,169)(134,170)(135,171)(136,172)(137,173)(138,174)(139,175)(140,176)(141,177)(142,178)(143,179)(144,180);
s1 := Sym(192)!( 1, 27)( 2, 28)( 3, 25)( 4, 26)( 5, 31)( 6, 32)( 7, 29)( 8, 30)( 9, 35)( 10, 36)( 11, 33)( 12, 34)( 13, 39)( 14, 40)( 15, 37)( 16, 38)( 17, 43)( 18, 44)( 19, 41)( 20, 42)( 21, 47)( 22, 48)( 23, 45)( 24, 46)( 49, 75)( 50, 76)( 51, 73)( 52, 74)( 53, 79)( 54, 80)( 55, 77)( 56, 78)( 57, 83)( 58, 84)( 59, 81)( 60, 82)( 61, 87)( 62, 88)( 63, 85)( 64, 86)( 65, 91)( 66, 92)( 67, 89)( 68, 90)( 69, 95)( 70, 96)( 71, 93)( 72, 94)( 97,123)( 98,124)( 99,121)(100,122)(101,127)(102,128)(103,125)(104,126)(105,131)(106,132)(107,129)(108,130)(109,135)(110,136)(111,133)(112,134)(113,139)(114,140)(115,137)(116,138)(117,143)(118,144)(119,141)(120,142)(145,171)(146,172)(147,169)(148,170)(149,175)(150,176)(151,173)(152,174)(153,179)(154,180)(155,177)(156,178)(157,183)(158,184)(159,181)(160,182)(161,187)(162,188)(163,185)(164,186)(165,191)(166,192)(167,189)(168,190);
s2 := Sym(192)!( 2, 3)( 5, 9)( 6, 11)( 7, 10)( 8, 12)( 14, 15)( 17, 21)( 18, 23)( 19, 22)( 20, 24)( 26, 27)( 29, 33)( 30, 35)( 31, 34)( 32, 36)( 38, 39)( 41, 45)( 42, 47)( 43, 46)( 44, 48)( 50, 51)( 53, 57)( 54, 59)( 55, 58)( 56, 60)( 62, 63)( 65, 69)( 66, 71)( 67, 70)( 68, 72)( 74, 75)( 77, 81)( 78, 83)( 79, 82)( 80, 84)( 86, 87)( 89, 93)( 90, 95)( 91, 94)( 92, 96)( 98, 99)(101,105)(102,107)(103,106)(104,108)(110,111)(113,117)(114,119)(115,118)(116,120)(122,123)(125,129)(126,131)(127,130)(128,132)(134,135)(137,141)(138,143)(139,142)(140,144)(146,147)(149,153)(150,155)(151,154)(152,156)(158,159)(161,165)(162,167)(163,166)(164,168)(170,171)(173,177)(174,179)(175,178)(176,180)(182,183)(185,189)(186,191)(187,190)(188,192);
s3 := Sym(192)!( 1,105)( 2,108)( 3,107)( 4,106)( 5,101)( 6,104)( 7,103)( 8,102)( 9, 97)( 10,100)( 11, 99)( 12, 98)( 13,117)( 14,120)( 15,119)( 16,118)( 17,113)( 18,116)( 19,115)( 20,114)( 21,109)( 22,112)( 23,111)( 24,110)( 25,129)( 26,132)( 27,131)( 28,130)( 29,125)( 30,128)( 31,127)( 32,126)( 33,121)( 34,124)( 35,123)( 36,122)( 37,141)( 38,144)( 39,143)( 40,142)( 41,137)( 42,140)( 43,139)( 44,138)( 45,133)( 46,136)( 47,135)( 48,134)( 49,153)( 50,156)( 51,155)( 52,154)( 53,149)( 54,152)( 55,151)( 56,150)( 57,145)( 58,148)( 59,147)( 60,146)( 61,165)( 62,168)( 63,167)( 64,166)( 65,161)( 66,164)( 67,163)( 68,162)( 69,157)( 70,160)( 71,159)( 72,158)( 73,177)( 74,180)( 75,179)( 76,178)( 77,173)( 78,176)( 79,175)( 80,174)( 81,169)( 82,172)( 83,171)( 84,170)( 85,189)( 86,192)( 87,191)( 88,190)( 89,185)( 90,188)( 91,187)( 92,186)( 93,181)( 94,184)( 95,183)( 96,182);
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, s0*s1*s2*s1*s0*s1*s2*s1,
s1*s2*s1*s2*s1*s2*s1*s2, s1*s2*s3*s2*s3*s2*s1*s2*s3*s2*s3*s2,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;
References : None.
to this polytope