Polytope of Type {4,6,4} with group SmallGroup(384,20163)

Overview

Group: SmallGroup(384,20163)
Rank: 4
Schläfli Type: {4,6,4}
Vertices, edges, …: 4, 24, 24, 8
Order of s₀s₁s₂s₃: 6
Order of s₀s₁s₂s₃s₂s₁: 2
Also known as: if this polytope has a name.

Special Properties

Non-Orientable
Flat

Quotients maximal quotients in bold

2-fold

4-fold

8-fold

{4,3,2}*48

Covers minimal covers in bold

2-fold

3-fold

5-fold

Irregular Quotients of which this is a minimal cover

Click an entry to reveal its facets and vertex figures.

P/N, where N=<(s2*s3)^2> of order 2

4 facets

4 of {4,6}*48b

4 vertex figures

4 of 2-fold non-regular quotient of {6,4}*96

Representations

Permutation Representation (GAP)

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

Permutation Representation (Magma)

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

References

None.

to this polytope.