Aliases: C12.11S4, Q8.4D18, C4○D4⋊1D9, Q8⋊D9⋊3C2, C6.23(C2×S4), Q8.D9⋊3C2, C3.(C4.6S4), C4.6(C3.S4), Q8.C18⋊2C2, (C3×Q8).12D6, Q8⋊C9.4C22, C2.9(C2×C3.S4), (C3×C4○D4).3S3, SmallGroup(288,339)
Series: Derived ►Chief ►Lower central ►Upper central
| Q8⋊C9 — C12.11S4 |
Generators and relations for C12.11S4
G = < a,b,c,d,e | a12=e2=1, b2=c2=a6, d3=a4, ab=ba, ac=ca, ad=da, eae=a5, cbc-1=a6b, dbd-1=a6bc, ebe=bc, dcd-1=b, ece=a6c, ede=a8d2 >
Subgroups: 359 in 65 conjugacy classes, 15 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, S3, C6, C6, C8, C2×C4, D4, Q8, Q8, C9, Dic3, C12, C12, D6, C2×C6, C2×C8, D8, SD16, Q16, C4○D4, C4○D4, D9, C18, C3⋊C8, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C3×D4, C3×Q8, C4○D8, Dic9, C36, D18, C2×C3⋊C8, D4⋊S3, D4.S3, Q8⋊2S3, C3⋊Q16, C4○D12, C3×C4○D4, Q8⋊C9, C4×D9, Q8.13D6, Q8.D9, Q8⋊D9, Q8.C18, C12.11S4
Quotients: C1, C2, C22, S3, D6, D9, S4, D18, C2×S4, C3.S4, C4.6S4, C2×C3.S4, C12.11S4
Character table of C12.11S4
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 6A | 6B | 8A | 8B | 8C | 8D | 9A | 9B | 9C | 12A | 12B | 12C | 18A | 18B | 18C | 36A | 36B | 36C | 36D | 36E | 36F | |
| size | 1 | 1 | 6 | 36 | 2 | 1 | 1 | 6 | 36 | 2 | 12 | 18 | 18 | 18 | 18 | 8 | 8 | 8 | 2 | 2 | 12 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ3 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ4 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ5 | 2 | 2 | 2 | 0 | 2 | 2 | 2 | 2 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 2 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ6 | 2 | 2 | -2 | 0 | 2 | -2 | -2 | 2 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -2 | -2 | 2 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ7 | 2 | 2 | 2 | 0 | -1 | 2 | 2 | 2 | 0 | -1 | -1 | 0 | 0 | 0 | 0 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | -1 | -1 | -1 | ζ98+ζ9 | ζ95+ζ94 | ζ97+ζ92 | ζ98+ζ9 | ζ98+ζ9 | ζ97+ζ92 | ζ97+ζ92 | ζ95+ζ94 | ζ95+ζ94 | orthogonal lifted from D9 |
| ρ8 | 2 | 2 | 2 | 0 | -1 | 2 | 2 | 2 | 0 | -1 | -1 | 0 | 0 | 0 | 0 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | -1 | -1 | -1 | ζ97+ζ92 | ζ98+ζ9 | ζ95+ζ94 | ζ97+ζ92 | ζ97+ζ92 | ζ95+ζ94 | ζ95+ζ94 | ζ98+ζ9 | ζ98+ζ9 | orthogonal lifted from D9 |
| ρ9 | 2 | 2 | -2 | 0 | -1 | -2 | -2 | 2 | 0 | -1 | 1 | 0 | 0 | 0 | 0 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | 1 | 1 | -1 | ζ98+ζ9 | ζ95+ζ94 | ζ97+ζ92 | -ζ98-ζ9 | -ζ98-ζ9 | -ζ97-ζ92 | -ζ97-ζ92 | -ζ95-ζ94 | -ζ95-ζ94 | orthogonal lifted from D18 |
| ρ10 | 2 | 2 | -2 | 0 | -1 | -2 | -2 | 2 | 0 | -1 | 1 | 0 | 0 | 0 | 0 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | 1 | 1 | -1 | ζ95+ζ94 | ζ97+ζ92 | ζ98+ζ9 | -ζ95-ζ94 | -ζ95-ζ94 | -ζ98-ζ9 | -ζ98-ζ9 | -ζ97-ζ92 | -ζ97-ζ92 | orthogonal lifted from D18 |
| ρ11 | 2 | 2 | 2 | 0 | -1 | 2 | 2 | 2 | 0 | -1 | -1 | 0 | 0 | 0 | 0 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | -1 | -1 | -1 | ζ95+ζ94 | ζ97+ζ92 | ζ98+ζ9 | ζ95+ζ94 | ζ95+ζ94 | ζ98+ζ9 | ζ98+ζ9 | ζ97+ζ92 | ζ97+ζ92 | orthogonal lifted from D9 |
| ρ12 | 2 | 2 | -2 | 0 | -1 | -2 | -2 | 2 | 0 | -1 | 1 | 0 | 0 | 0 | 0 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | 1 | 1 | -1 | ζ97+ζ92 | ζ98+ζ9 | ζ95+ζ94 | -ζ97-ζ92 | -ζ97-ζ92 | -ζ95-ζ94 | -ζ95-ζ94 | -ζ98-ζ9 | -ζ98-ζ9 | orthogonal lifted from D18 |
| ρ13 | 2 | -2 | 0 | 0 | 2 | 2i | -2i | 0 | 0 | -2 | 0 | -√2 | -√-2 | √2 | √-2 | -1 | -1 | -1 | -2i | 2i | 0 | 1 | 1 | 1 | -i | i | -i | i | -i | i | complex lifted from C4.6S4 |
| ρ14 | 2 | -2 | 0 | 0 | 2 | -2i | 2i | 0 | 0 | -2 | 0 | -√2 | √-2 | √2 | -√-2 | -1 | -1 | -1 | 2i | -2i | 0 | 1 | 1 | 1 | i | -i | i | -i | i | -i | complex lifted from C4.6S4 |
| ρ15 | 2 | -2 | 0 | 0 | 2 | -2i | 2i | 0 | 0 | -2 | 0 | √2 | -√-2 | -√2 | √-2 | -1 | -1 | -1 | 2i | -2i | 0 | 1 | 1 | 1 | i | -i | i | -i | i | -i | complex lifted from C4.6S4 |
| ρ16 | 2 | -2 | 0 | 0 | 2 | 2i | -2i | 0 | 0 | -2 | 0 | √2 | √-2 | -√2 | -√-2 | -1 | -1 | -1 | -2i | 2i | 0 | 1 | 1 | 1 | -i | i | -i | i | -i | i | complex lifted from C4.6S4 |
| ρ17 | 3 | 3 | 1 | 1 | 3 | -3 | -3 | -1 | -1 | 3 | 1 | 1 | -1 | 1 | -1 | 0 | 0 | 0 | -3 | -3 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×S4 |
| ρ18 | 3 | 3 | -1 | -1 | 3 | 3 | 3 | -1 | -1 | 3 | -1 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 3 | 3 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S4 |
| ρ19 | 3 | 3 | -1 | 1 | 3 | 3 | 3 | -1 | 1 | 3 | -1 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 3 | 3 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S4 |
| ρ20 | 3 | 3 | 1 | -1 | 3 | -3 | -3 | -1 | 1 | 3 | 1 | -1 | 1 | -1 | 1 | 0 | 0 | 0 | -3 | -3 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×S4 |
| ρ21 | 4 | -4 | 0 | 0 | 4 | 4i | -4i | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | -4i | 4i | 0 | -1 | -1 | -1 | i | -i | i | -i | i | -i | complex lifted from C4.6S4 |
| ρ22 | 4 | -4 | 0 | 0 | 4 | -4i | 4i | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 4i | -4i | 0 | -1 | -1 | -1 | -i | i | -i | i | -i | i | complex lifted from C4.6S4 |
| ρ23 | 4 | -4 | 0 | 0 | -2 | 4i | -4i | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | -ζ95-ζ94 | -ζ98-ζ9 | -ζ97-ζ92 | 2i | -2i | 0 | ζ95+ζ94 | ζ97+ζ92 | ζ98+ζ9 | ζ43ζ95+ζ43ζ94 | ζ4ζ95+ζ4ζ94 | ζ43ζ98+ζ43ζ9 | ζ4ζ98+ζ4ζ9 | ζ43ζ97+ζ43ζ92 | ζ4ζ97+ζ4ζ92 | complex faithful |
| ρ24 | 4 | -4 | 0 | 0 | -2 | 4i | -4i | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | -ζ98-ζ9 | -ζ97-ζ92 | -ζ95-ζ94 | 2i | -2i | 0 | ζ98+ζ9 | ζ95+ζ94 | ζ97+ζ92 | ζ43ζ98+ζ43ζ9 | ζ4ζ98+ζ4ζ9 | ζ43ζ97+ζ43ζ92 | ζ4ζ97+ζ4ζ92 | ζ43ζ95+ζ43ζ94 | ζ4ζ95+ζ4ζ94 | complex faithful |
| ρ25 | 4 | -4 | 0 | 0 | -2 | -4i | 4i | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | -ζ97-ζ92 | -ζ95-ζ94 | -ζ98-ζ9 | -2i | 2i | 0 | ζ97+ζ92 | ζ98+ζ9 | ζ95+ζ94 | ζ4ζ97+ζ4ζ92 | ζ43ζ97+ζ43ζ92 | ζ4ζ95+ζ4ζ94 | ζ43ζ95+ζ43ζ94 | ζ4ζ98+ζ4ζ9 | ζ43ζ98+ζ43ζ9 | complex faithful |
| ρ26 | 4 | -4 | 0 | 0 | -2 | -4i | 4i | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | -ζ98-ζ9 | -ζ97-ζ92 | -ζ95-ζ94 | -2i | 2i | 0 | ζ98+ζ9 | ζ95+ζ94 | ζ97+ζ92 | ζ4ζ98+ζ4ζ9 | ζ43ζ98+ζ43ζ9 | ζ4ζ97+ζ4ζ92 | ζ43ζ97+ζ43ζ92 | ζ4ζ95+ζ4ζ94 | ζ43ζ95+ζ43ζ94 | complex faithful |
| ρ27 | 4 | -4 | 0 | 0 | -2 | 4i | -4i | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | -ζ97-ζ92 | -ζ95-ζ94 | -ζ98-ζ9 | 2i | -2i | 0 | ζ97+ζ92 | ζ98+ζ9 | ζ95+ζ94 | ζ43ζ97+ζ43ζ92 | ζ4ζ97+ζ4ζ92 | ζ43ζ95+ζ43ζ94 | ζ4ζ95+ζ4ζ94 | ζ43ζ98+ζ43ζ9 | ζ4ζ98+ζ4ζ9 | complex faithful |
| ρ28 | 4 | -4 | 0 | 0 | -2 | -4i | 4i | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | -ζ95-ζ94 | -ζ98-ζ9 | -ζ97-ζ92 | -2i | 2i | 0 | ζ95+ζ94 | ζ97+ζ92 | ζ98+ζ9 | ζ4ζ95+ζ4ζ94 | ζ43ζ95+ζ43ζ94 | ζ4ζ98+ζ4ζ9 | ζ43ζ98+ζ43ζ9 | ζ4ζ97+ζ4ζ92 | ζ43ζ97+ζ43ζ92 | complex faithful |
| ρ29 | 6 | 6 | 2 | 0 | -3 | -6 | -6 | -2 | 0 | -3 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 3 | 3 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×C3.S4 |
| ρ30 | 6 | 6 | -2 | 0 | -3 | 6 | 6 | -2 | 0 | -3 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -3 | -3 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C3.S4 |
►On 144 points
Generators in S144
(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)
(1 27 7 33)(2 28 8 34)(3 29 9 35)(4 30 10 36)(5 31 11 25)(6 32 12 26)(13 80 19 74)(14 81 20 75)(15 82 21 76)(16 83 22 77)(17 84 23 78)(18 73 24 79)(37 119 43 113)(38 120 44 114)(39 109 45 115)(40 110 46 116)(41 111 47 117)(42 112 48 118)(49 129 55 123)(50 130 56 124)(51 131 57 125)(52 132 58 126)(53 121 59 127)(54 122 60 128)(61 87 67 93)(62 88 68 94)(63 89 69 95)(64 90 70 96)(65 91 71 85)(66 92 72 86)(97 138 103 144)(98 139 104 133)(99 140 105 134)(100 141 106 135)(101 142 107 136)(102 143 108 137)
(1 87 7 93)(2 88 8 94)(3 89 9 95)(4 90 10 96)(5 91 11 85)(6 92 12 86)(13 50 19 56)(14 51 20 57)(15 52 21 58)(16 53 22 59)(17 54 23 60)(18 55 24 49)(25 71 31 65)(26 72 32 66)(27 61 33 67)(28 62 34 68)(29 63 35 69)(30 64 36 70)(37 139 43 133)(38 140 44 134)(39 141 45 135)(40 142 46 136)(41 143 47 137)(42 144 48 138)(73 129 79 123)(74 130 80 124)(75 131 81 125)(76 132 82 126)(77 121 83 127)(78 122 84 128)(97 112 103 118)(98 113 104 119)(99 114 105 120)(100 115 106 109)(101 116 107 110)(102 117 108 111)
(1 23 139 5 15 143 9 19 135)(2 24 140 6 16 144 10 20 136)(3 13 141 7 17 133 11 21 137)(4 14 142 8 18 134 12 22 138)(25 58 111 29 50 115 33 54 119)(26 59 112 30 51 116 34 55 120)(27 60 113 31 52 117 35 56 109)(28 49 114 32 53 118 36 57 110)(37 65 76 41 69 80 45 61 84)(38 66 77 42 70 81 46 62 73)(39 67 78 43 71 82 47 63 74)(40 68 79 44 72 83 48 64 75)(85 126 102 89 130 106 93 122 98)(86 127 103 90 131 107 94 123 99)(87 128 104 91 132 108 95 124 100)(88 129 105 92 121 97 96 125 101)
(2 6)(3 11)(5 9)(8 12)(13 133)(14 138)(15 143)(16 136)(17 141)(18 134)(19 139)(20 144)(21 137)(22 142)(23 135)(24 140)(25 63)(26 68)(27 61)(28 66)(29 71)(30 64)(31 69)(32 62)(33 67)(34 72)(35 65)(36 70)(37 56)(38 49)(39 54)(40 59)(41 52)(42 57)(43 50)(44 55)(45 60)(46 53)(47 58)(48 51)(73 114)(74 119)(75 112)(76 117)(77 110)(78 115)(79 120)(80 113)(81 118)(82 111)(83 116)(84 109)(85 95)(86 88)(87 93)(89 91)(90 96)(92 94)(97 131)(98 124)(99 129)(100 122)(101 127)(102 132)(103 125)(104 130)(105 123)(106 128)(107 121)(108 126)
G:=sub<Sym(144)| (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), (1,27,7,33)(2,28,8,34)(3,29,9,35)(4,30,10,36)(5,31,11,25)(6,32,12,26)(13,80,19,74)(14,81,20,75)(15,82,21,76)(16,83,22,77)(17,84,23,78)(18,73,24,79)(37,119,43,113)(38,120,44,114)(39,109,45,115)(40,110,46,116)(41,111,47,117)(42,112,48,118)(49,129,55,123)(50,130,56,124)(51,131,57,125)(52,132,58,126)(53,121,59,127)(54,122,60,128)(61,87,67,93)(62,88,68,94)(63,89,69,95)(64,90,70,96)(65,91,71,85)(66,92,72,86)(97,138,103,144)(98,139,104,133)(99,140,105,134)(100,141,106,135)(101,142,107,136)(102,143,108,137), (1,87,7,93)(2,88,8,94)(3,89,9,95)(4,90,10,96)(5,91,11,85)(6,92,12,86)(13,50,19,56)(14,51,20,57)(15,52,21,58)(16,53,22,59)(17,54,23,60)(18,55,24,49)(25,71,31,65)(26,72,32,66)(27,61,33,67)(28,62,34,68)(29,63,35,69)(30,64,36,70)(37,139,43,133)(38,140,44,134)(39,141,45,135)(40,142,46,136)(41,143,47,137)(42,144,48,138)(73,129,79,123)(74,130,80,124)(75,131,81,125)(76,132,82,126)(77,121,83,127)(78,122,84,128)(97,112,103,118)(98,113,104,119)(99,114,105,120)(100,115,106,109)(101,116,107,110)(102,117,108,111), (1,23,139,5,15,143,9,19,135)(2,24,140,6,16,144,10,20,136)(3,13,141,7,17,133,11,21,137)(4,14,142,8,18,134,12,22,138)(25,58,111,29,50,115,33,54,119)(26,59,112,30,51,116,34,55,120)(27,60,113,31,52,117,35,56,109)(28,49,114,32,53,118,36,57,110)(37,65,76,41,69,80,45,61,84)(38,66,77,42,70,81,46,62,73)(39,67,78,43,71,82,47,63,74)(40,68,79,44,72,83,48,64,75)(85,126,102,89,130,106,93,122,98)(86,127,103,90,131,107,94,123,99)(87,128,104,91,132,108,95,124,100)(88,129,105,92,121,97,96,125,101), (2,6)(3,11)(5,9)(8,12)(13,133)(14,138)(15,143)(16,136)(17,141)(18,134)(19,139)(20,144)(21,137)(22,142)(23,135)(24,140)(25,63)(26,68)(27,61)(28,66)(29,71)(30,64)(31,69)(32,62)(33,67)(34,72)(35,65)(36,70)(37,56)(38,49)(39,54)(40,59)(41,52)(42,57)(43,50)(44,55)(45,60)(46,53)(47,58)(48,51)(73,114)(74,119)(75,112)(76,117)(77,110)(78,115)(79,120)(80,113)(81,118)(82,111)(83,116)(84,109)(85,95)(86,88)(87,93)(89,91)(90,96)(92,94)(97,131)(98,124)(99,129)(100,122)(101,127)(102,132)(103,125)(104,130)(105,123)(106,128)(107,121)(108,126)>;
G:=Group( (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), (1,27,7,33)(2,28,8,34)(3,29,9,35)(4,30,10,36)(5,31,11,25)(6,32,12,26)(13,80,19,74)(14,81,20,75)(15,82,21,76)(16,83,22,77)(17,84,23,78)(18,73,24,79)(37,119,43,113)(38,120,44,114)(39,109,45,115)(40,110,46,116)(41,111,47,117)(42,112,48,118)(49,129,55,123)(50,130,56,124)(51,131,57,125)(52,132,58,126)(53,121,59,127)(54,122,60,128)(61,87,67,93)(62,88,68,94)(63,89,69,95)(64,90,70,96)(65,91,71,85)(66,92,72,86)(97,138,103,144)(98,139,104,133)(99,140,105,134)(100,141,106,135)(101,142,107,136)(102,143,108,137), (1,87,7,93)(2,88,8,94)(3,89,9,95)(4,90,10,96)(5,91,11,85)(6,92,12,86)(13,50,19,56)(14,51,20,57)(15,52,21,58)(16,53,22,59)(17,54,23,60)(18,55,24,49)(25,71,31,65)(26,72,32,66)(27,61,33,67)(28,62,34,68)(29,63,35,69)(30,64,36,70)(37,139,43,133)(38,140,44,134)(39,141,45,135)(40,142,46,136)(41,143,47,137)(42,144,48,138)(73,129,79,123)(74,130,80,124)(75,131,81,125)(76,132,82,126)(77,121,83,127)(78,122,84,128)(97,112,103,118)(98,113,104,119)(99,114,105,120)(100,115,106,109)(101,116,107,110)(102,117,108,111), (1,23,139,5,15,143,9,19,135)(2,24,140,6,16,144,10,20,136)(3,13,141,7,17,133,11,21,137)(4,14,142,8,18,134,12,22,138)(25,58,111,29,50,115,33,54,119)(26,59,112,30,51,116,34,55,120)(27,60,113,31,52,117,35,56,109)(28,49,114,32,53,118,36,57,110)(37,65,76,41,69,80,45,61,84)(38,66,77,42,70,81,46,62,73)(39,67,78,43,71,82,47,63,74)(40,68,79,44,72,83,48,64,75)(85,126,102,89,130,106,93,122,98)(86,127,103,90,131,107,94,123,99)(87,128,104,91,132,108,95,124,100)(88,129,105,92,121,97,96,125,101), (2,6)(3,11)(5,9)(8,12)(13,133)(14,138)(15,143)(16,136)(17,141)(18,134)(19,139)(20,144)(21,137)(22,142)(23,135)(24,140)(25,63)(26,68)(27,61)(28,66)(29,71)(30,64)(31,69)(32,62)(33,67)(34,72)(35,65)(36,70)(37,56)(38,49)(39,54)(40,59)(41,52)(42,57)(43,50)(44,55)(45,60)(46,53)(47,58)(48,51)(73,114)(74,119)(75,112)(76,117)(77,110)(78,115)(79,120)(80,113)(81,118)(82,111)(83,116)(84,109)(85,95)(86,88)(87,93)(89,91)(90,96)(92,94)(97,131)(98,124)(99,129)(100,122)(101,127)(102,132)(103,125)(104,130)(105,123)(106,128)(107,121)(108,126) );
G=PermutationGroup([[(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)], [(1,27,7,33),(2,28,8,34),(3,29,9,35),(4,30,10,36),(5,31,11,25),(6,32,12,26),(13,80,19,74),(14,81,20,75),(15,82,21,76),(16,83,22,77),(17,84,23,78),(18,73,24,79),(37,119,43,113),(38,120,44,114),(39,109,45,115),(40,110,46,116),(41,111,47,117),(42,112,48,118),(49,129,55,123),(50,130,56,124),(51,131,57,125),(52,132,58,126),(53,121,59,127),(54,122,60,128),(61,87,67,93),(62,88,68,94),(63,89,69,95),(64,90,70,96),(65,91,71,85),(66,92,72,86),(97,138,103,144),(98,139,104,133),(99,140,105,134),(100,141,106,135),(101,142,107,136),(102,143,108,137)], [(1,87,7,93),(2,88,8,94),(3,89,9,95),(4,90,10,96),(5,91,11,85),(6,92,12,86),(13,50,19,56),(14,51,20,57),(15,52,21,58),(16,53,22,59),(17,54,23,60),(18,55,24,49),(25,71,31,65),(26,72,32,66),(27,61,33,67),(28,62,34,68),(29,63,35,69),(30,64,36,70),(37,139,43,133),(38,140,44,134),(39,141,45,135),(40,142,46,136),(41,143,47,137),(42,144,48,138),(73,129,79,123),(74,130,80,124),(75,131,81,125),(76,132,82,126),(77,121,83,127),(78,122,84,128),(97,112,103,118),(98,113,104,119),(99,114,105,120),(100,115,106,109),(101,116,107,110),(102,117,108,111)], [(1,23,139,5,15,143,9,19,135),(2,24,140,6,16,144,10,20,136),(3,13,141,7,17,133,11,21,137),(4,14,142,8,18,134,12,22,138),(25,58,111,29,50,115,33,54,119),(26,59,112,30,51,116,34,55,120),(27,60,113,31,52,117,35,56,109),(28,49,114,32,53,118,36,57,110),(37,65,76,41,69,80,45,61,84),(38,66,77,42,70,81,46,62,73),(39,67,78,43,71,82,47,63,74),(40,68,79,44,72,83,48,64,75),(85,126,102,89,130,106,93,122,98),(86,127,103,90,131,107,94,123,99),(87,128,104,91,132,108,95,124,100),(88,129,105,92,121,97,96,125,101)], [(2,6),(3,11),(5,9),(8,12),(13,133),(14,138),(15,143),(16,136),(17,141),(18,134),(19,139),(20,144),(21,137),(22,142),(23,135),(24,140),(25,63),(26,68),(27,61),(28,66),(29,71),(30,64),(31,69),(32,62),(33,67),(34,72),(35,65),(36,70),(37,56),(38,49),(39,54),(40,59),(41,52),(42,57),(43,50),(44,55),(45,60),(46,53),(47,58),(48,51),(73,114),(74,119),(75,112),(76,117),(77,110),(78,115),(79,120),(80,113),(81,118),(82,111),(83,116),(84,109),(85,95),(86,88),(87,93),(89,91),(90,96),(92,94),(97,131),(98,124),(99,129),(100,122),(101,127),(102,132),(103,125),(104,130),(105,123),(106,128),(107,121),(108,126)]])
Matrix representation of C12.11S4 ►in GL4(𝔽73) generated by
| 46 | 0 | 0 | 0 |
| 0 | 46 | 0 | 0 |
| 0 | 0 | 0 | 72 |
| 0 | 0 | 1 | 1 |
| 61 | 72 | 0 | 0 |
| 72 | 12 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 61 | 0 | 0 |
| 61 | 72 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 5 | 7 | 0 | 0 |
| 6 | 67 | 0 | 0 |
| 0 | 0 | 70 | 28 |
| 0 | 0 | 45 | 42 |
| 1 | 0 | 0 | 0 |
| 61 | 72 | 0 | 0 |
| 0 | 0 | 70 | 42 |
| 0 | 0 | 45 | 3 |
G:=sub<GL(4,GF(73))| [46,0,0,0,0,46,0,0,0,0,0,1,0,0,72,1],[61,72,0,0,72,12,0,0,0,0,1,0,0,0,0,1],[1,61,0,0,61,72,0,0,0,0,1,0,0,0,0,1],[5,6,0,0,7,67,0,0,0,0,70,45,0,0,28,42],[1,61,0,0,0,72,0,0,0,0,70,45,0,0,42,3] >;
C12.11S4 in GAP, Magma, Sage, TeX
C_{12}._{11}S_4 % in TeX
G:=Group("C12.11S4"); // GroupNames label
G:=SmallGroup(288,339);
// by ID
G=gap.SmallGroup(288,339);
# by ID
G:=PCGroup([7,-2,-2,-3,-3,-2,2,-2,1016,422,142,675,2524,1908,172,1517,1153,285,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^12=e^2=1,b^2=c^2=a^6,d^3=a^4,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e=a^5,c*b*c^-1=a^6*b,d*b*d^-1=a^6*b*c,e*b*e=b*c,d*c*d^-1=b,e*c*e=a^6*c,e*d*e=a^8*d^2>;
// generators/relations
Export