Polytope of Type {8,12,4} with group SmallGroup(768,1087796)

Overview

Group: SmallGroup(768,1087796)
Rank: 4
Schläfli Type: {8,12,4}
Vertices, edges, …: 8, 48, 24, 4
Order of s₀s₁s₂s₃: 24
Order of s₀s₁s₂s₃s₂s₁: 2
Also known as: if this polytope has a name.

Special Properties

Universal
Non-Orientable
Flat

Quotients maximal quotients in bold

2-fold

{4,12,4}*384b

4-fold

8-fold

{2,6,4}*96c

16-fold

{2,3,4}*48

Covers minimal covers in bold

None in this atlas.

Irregular Quotients of which this is a minimal cover

None.

Representations

Permutation Representation (GAP)

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

Permutation Representation (Magma)

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

References