Polytope of Type {8,6,4} with group SmallGroup(384,12600)

Overview

Group: SmallGroup(384,12600)
Rank: 4
Schläfli Type: {8,6,4}
Vertices, edges, …: 8, 24, 12, 4
Order of s₀s₁s₂s₃: 24
Order of s₀s₁s₂s₃s₂s₁: 2
Also known as: {{8,6|2},{6,4|2}}. if this polytope has another name.

Special Properties

Universal
Orientable
Flat

Quotients maximal quotients in bold

2-fold

3-fold

{8,2,4}*128

4-fold

6-fold

8-fold

{2,6,2}*48

12-fold

16-fold

{2,3,2}*24

24-fold

{2,2,2}*16

Covers minimal covers in bold

2-fold

3-fold

5-fold

Irregular Quotients of which this is a minimal cover

None.

Representations

Permutation Representation (GAP)

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

Permutation Representation (Magma)

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

References