Part of the Atlas of Small Regular Polytopes

Polytope of Type {4,6,4}

Atlas Canonical Name {4,6,4}*768k

Overview

Group
SmallGroup(768,1090220)
Rank
4
Schläfli Type
{4,6,4}
Vertices, edges, …
16, 48, 48, 4
Order of s0s1s2s3
6
Order of s0s1s2s3s2s1
4
Also known as
if this polytope has a name.

Special Properties

  • Universal
  • Non-Orientable
  • Flat

Quotients maximal quotients in bold

2-fold

4-fold

8-fold

16-fold

Covers minimal covers in bold

None in this atlas.

Irregular Quotients of which this is a minimal cover

Click an entry to reveal its facets and vertex figures.

P/N, where N=<s0*(s2*s1)^2*s0*(s1*s2)^2> of order 2

4 facets

8 vertex figures

P/N, where N=<(s0*s1)^2*s2*s1*s0*s1*s2> of order 2

4 facets

8 vertex figures

P/N, where N=<(s1*s2)^3> of order 2

4 facets

12 vertex figures

P/N, where N=<(s0*s1*s2*s1)^2, (s1*s0*s1*s2)^2> of order 4

4 facets

4 vertex figures

Representations

Permutation Representation (GAP) 
s0 := (  1,  7)(  2,  8)(  3,  5)(  4,  6)(  9, 15)( 10, 16)( 11, 13)( 12, 14)( 17, 23)( 18, 24)( 19, 21)( 20, 22)( 25, 31)( 26, 32)( 27, 29)( 28, 30)( 33, 39)( 34, 40)( 35, 37)( 36, 38)( 41, 47)( 42, 48)( 43, 45)( 44, 46)( 49, 55)( 50, 56)( 51, 53)( 52, 54)( 57, 63)( 58, 64)( 59, 61)( 60, 62)( 65, 71)( 66, 72)( 67, 69)( 68, 70)( 73, 79)( 74, 80)( 75, 77)( 76, 78)( 81, 87)( 82, 88)( 83, 85)( 84, 86)( 89, 95)( 90, 96)( 91, 93)( 92, 94)( 97,104)( 98,103)( 99,102)(100,101)(105,112)(106,111)(107,110)(108,109)(113,120)(114,119)(115,118)(116,117)(121,128)(122,127)(123,126)(124,125)(129,136)(130,135)(131,134)(132,133)(137,144)(138,143)(139,142)(140,141)(145,152)(146,151)(147,150)(148,149)(153,160)(154,159)(155,158)(156,157)(161,168)(162,167)(163,166)(164,165)(169,176)(170,175)(171,174)(172,173)(177,184)(178,183)(179,182)(180,181)(185,192)(186,191)(187,190)(188,189);;
s1 := (  1, 97)(  2, 98)(  3, 99)(  4,100)(  5,103)(  6,104)(  7,101)(  8,102)(  9,105)( 10,106)( 11,107)( 12,108)( 13,111)( 14,112)( 15,109)( 16,110)( 17,121)( 18,122)( 19,123)( 20,124)( 21,127)( 22,128)( 23,125)( 24,126)( 25,113)( 26,114)( 27,115)( 28,116)( 29,119)( 30,120)( 31,117)( 32,118)( 33,161)( 34,162)( 35,163)( 36,164)( 37,167)( 38,168)( 39,165)( 40,166)( 41,169)( 42,170)( 43,171)( 44,172)( 45,175)( 46,176)( 47,173)( 48,174)( 49,185)( 50,186)( 51,187)( 52,188)( 53,191)( 54,192)( 55,189)( 56,190)( 57,177)( 58,178)( 59,179)( 60,180)( 61,183)( 62,184)( 63,181)( 64,182)( 65,129)( 66,130)( 67,131)( 68,132)( 69,135)( 70,136)( 71,133)( 72,134)( 73,137)( 74,138)( 75,139)( 76,140)( 77,143)( 78,144)( 79,141)( 80,142)( 81,153)( 82,154)( 83,155)( 84,156)( 85,159)( 86,160)( 87,157)( 88,158)( 89,145)( 90,146)( 91,147)( 92,148)( 93,151)( 94,152)( 95,149)( 96,150);;
s2 := (  1, 65)(  2, 66)(  3, 69)(  4, 70)(  5, 67)(  6, 68)(  7, 71)(  8, 72)(  9, 81)( 10, 82)( 11, 85)( 12, 86)( 13, 83)( 14, 84)( 15, 87)( 16, 88)( 17, 73)( 18, 74)( 19, 77)( 20, 78)( 21, 75)( 22, 76)( 23, 79)( 24, 80)( 25, 89)( 26, 90)( 27, 93)( 28, 94)( 29, 91)( 30, 92)( 31, 95)( 32, 96)( 35, 37)( 36, 38)( 41, 49)( 42, 50)( 43, 53)( 44, 54)( 45, 51)( 46, 52)( 47, 55)( 48, 56)( 59, 61)( 60, 62)( 97,161)( 98,162)( 99,165)(100,166)(101,163)(102,164)(103,167)(104,168)(105,177)(106,178)(107,181)(108,182)(109,179)(110,180)(111,183)(112,184)(113,169)(114,170)(115,173)(116,174)(117,171)(118,172)(119,175)(120,176)(121,185)(122,186)(123,189)(124,190)(125,187)(126,188)(127,191)(128,192)(131,133)(132,134)(137,145)(138,146)(139,149)(140,150)(141,147)(142,148)(143,151)(144,152)(155,157)(156,158);;
s3 := (  1,105)(  2,106)(  3,107)(  4,108)(  5,110)(  6,109)(  7,112)(  8,111)(  9, 97)( 10, 98)( 11, 99)( 12,100)( 13,102)( 14,101)( 15,104)( 16,103)( 17,121)( 18,122)( 19,123)( 20,124)( 21,126)( 22,125)( 23,128)( 24,127)( 25,113)( 26,114)( 27,115)( 28,116)( 29,118)( 30,117)( 31,120)( 32,119)( 33,137)( 34,138)( 35,139)( 36,140)( 37,142)( 38,141)( 39,144)( 40,143)( 41,129)( 42,130)( 43,131)( 44,132)( 45,134)( 46,133)( 47,136)( 48,135)( 49,153)( 50,154)( 51,155)( 52,156)( 53,158)( 54,157)( 55,160)( 56,159)( 57,145)( 58,146)( 59,147)( 60,148)( 61,150)( 62,149)( 63,152)( 64,151)( 65,169)( 66,170)( 67,171)( 68,172)( 69,174)( 70,173)( 71,176)( 72,175)( 73,161)( 74,162)( 75,163)( 76,164)( 77,166)( 78,165)( 79,168)( 80,167)( 81,185)( 82,186)( 83,187)( 84,188)( 85,190)( 86,189)( 87,192)( 88,191)( 89,177)( 90,178)( 91,179)( 92,180)( 93,182)( 94,181)( 95,184)( 96,183);;
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, s1*s2*s3*s1*s2*s3*s1*s2*s3, 
s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) 
s0 := Sym(192)!(  1,  7)(  2,  8)(  3,  5)(  4,  6)(  9, 15)( 10, 16)( 11, 13)( 12, 14)( 17, 23)( 18, 24)( 19, 21)( 20, 22)( 25, 31)( 26, 32)( 27, 29)( 28, 30)( 33, 39)( 34, 40)( 35, 37)( 36, 38)( 41, 47)( 42, 48)( 43, 45)( 44, 46)( 49, 55)( 50, 56)( 51, 53)( 52, 54)( 57, 63)( 58, 64)( 59, 61)( 60, 62)( 65, 71)( 66, 72)( 67, 69)( 68, 70)( 73, 79)( 74, 80)( 75, 77)( 76, 78)( 81, 87)( 82, 88)( 83, 85)( 84, 86)( 89, 95)( 90, 96)( 91, 93)( 92, 94)( 97,104)( 98,103)( 99,102)(100,101)(105,112)(106,111)(107,110)(108,109)(113,120)(114,119)(115,118)(116,117)(121,128)(122,127)(123,126)(124,125)(129,136)(130,135)(131,134)(132,133)(137,144)(138,143)(139,142)(140,141)(145,152)(146,151)(147,150)(148,149)(153,160)(154,159)(155,158)(156,157)(161,168)(162,167)(163,166)(164,165)(169,176)(170,175)(171,174)(172,173)(177,184)(178,183)(179,182)(180,181)(185,192)(186,191)(187,190)(188,189);
s1 := Sym(192)!(  1, 97)(  2, 98)(  3, 99)(  4,100)(  5,103)(  6,104)(  7,101)(  8,102)(  9,105)( 10,106)( 11,107)( 12,108)( 13,111)( 14,112)( 15,109)( 16,110)( 17,121)( 18,122)( 19,123)( 20,124)( 21,127)( 22,128)( 23,125)( 24,126)( 25,113)( 26,114)( 27,115)( 28,116)( 29,119)( 30,120)( 31,117)( 32,118)( 33,161)( 34,162)( 35,163)( 36,164)( 37,167)( 38,168)( 39,165)( 40,166)( 41,169)( 42,170)( 43,171)( 44,172)( 45,175)( 46,176)( 47,173)( 48,174)( 49,185)( 50,186)( 51,187)( 52,188)( 53,191)( 54,192)( 55,189)( 56,190)( 57,177)( 58,178)( 59,179)( 60,180)( 61,183)( 62,184)( 63,181)( 64,182)( 65,129)( 66,130)( 67,131)( 68,132)( 69,135)( 70,136)( 71,133)( 72,134)( 73,137)( 74,138)( 75,139)( 76,140)( 77,143)( 78,144)( 79,141)( 80,142)( 81,153)( 82,154)( 83,155)( 84,156)( 85,159)( 86,160)( 87,157)( 88,158)( 89,145)( 90,146)( 91,147)( 92,148)( 93,151)( 94,152)( 95,149)( 96,150);
s2 := Sym(192)!(  1, 65)(  2, 66)(  3, 69)(  4, 70)(  5, 67)(  6, 68)(  7, 71)(  8, 72)(  9, 81)( 10, 82)( 11, 85)( 12, 86)( 13, 83)( 14, 84)( 15, 87)( 16, 88)( 17, 73)( 18, 74)( 19, 77)( 20, 78)( 21, 75)( 22, 76)( 23, 79)( 24, 80)( 25, 89)( 26, 90)( 27, 93)( 28, 94)( 29, 91)( 30, 92)( 31, 95)( 32, 96)( 35, 37)( 36, 38)( 41, 49)( 42, 50)( 43, 53)( 44, 54)( 45, 51)( 46, 52)( 47, 55)( 48, 56)( 59, 61)( 60, 62)( 97,161)( 98,162)( 99,165)(100,166)(101,163)(102,164)(103,167)(104,168)(105,177)(106,178)(107,181)(108,182)(109,179)(110,180)(111,183)(112,184)(113,169)(114,170)(115,173)(116,174)(117,171)(118,172)(119,175)(120,176)(121,185)(122,186)(123,189)(124,190)(125,187)(126,188)(127,191)(128,192)(131,133)(132,134)(137,145)(138,146)(139,149)(140,150)(141,147)(142,148)(143,151)(144,152)(155,157)(156,158);
s3 := Sym(192)!(  1,105)(  2,106)(  3,107)(  4,108)(  5,110)(  6,109)(  7,112)(  8,111)(  9, 97)( 10, 98)( 11, 99)( 12,100)( 13,102)( 14,101)( 15,104)( 16,103)( 17,121)( 18,122)( 19,123)( 20,124)( 21,126)( 22,125)( 23,128)( 24,127)( 25,113)( 26,114)( 27,115)( 28,116)( 29,118)( 30,117)( 31,120)( 32,119)( 33,137)( 34,138)( 35,139)( 36,140)( 37,142)( 38,141)( 39,144)( 40,143)( 41,129)( 42,130)( 43,131)( 44,132)( 45,134)( 46,133)( 47,136)( 48,135)( 49,153)( 50,154)( 51,155)( 52,156)( 53,158)( 54,157)( 55,160)( 56,159)( 57,145)( 58,146)( 59,147)( 60,148)( 61,150)( 62,149)( 63,152)( 64,151)( 65,169)( 66,170)( 67,171)( 68,172)( 69,174)( 70,173)( 71,176)( 72,175)( 73,161)( 74,162)( 75,163)( 76,164)( 77,166)( 78,165)( 79,168)( 80,167)( 81,185)( 82,186)( 83,187)( 84,188)( 85,190)( 86,189)( 87,192)( 88,191)( 89,177)( 90,178)( 91,179)( 92,180)( 93,182)( 94,181)( 95,184)( 96,183);
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, 
s1*s2*s3*s1*s2*s3*s1*s2*s3, s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1 >;

References

None.

to this polytope.