Polytope of Type {28,4,6} with group SmallGroup(1344,7765)

Overview

Group: SmallGroup(1344,7765)
Rank: 4
Schläfli Type: {28,4,6}
Vertices, edges, …: 28, 56, 12, 6
Order of s₀s₁s₂s₃: 84
Order of s₀s₁s₂s₃s₂s₁: 2
Also known as: {{28,4|2},{4,6|2}}. if this polytope has another name.

Special Properties

Universal
Orientable
Flat

Quotients maximal quotients in bold

2-fold

3-fold

{28,4,2}*448

4-fold

6-fold

7-fold

{4,4,6}*192

8-fold

12-fold

{14,2,2}*112

14-fold

16-fold

{7,2,3}*84

21-fold

{4,4,2}*64

24-fold

{7,2,2}*56

28-fold

42-fold

56-fold

{2,2,3}*24

84-fold

{2,2,2}*16

Covers minimal covers in bold

None in this atlas.

Irregular Quotients of which this is a minimal cover

None.

Representations

Permutation Representation (GAP)

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

Permutation Representation (Magma)

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

References