Aliases: C12.3S4, Q8.3D18, C6.22(C2×S4), C4○D4.2D9, Q8.D9⋊2C2, C4.2(C3.S4), C3.(C4.S4), (C3×Q8).11D6, Q8⋊C9.3C22, Q8.C18.1C2, C2.8(C2×C3.S4), (C3×C4○D4).2S3, SmallGroup(288,338)
Series: Derived ►Chief ►Lower central ►Upper central
| Q8⋊C9 — C12.3S4 |
Generators and relations for C12.3S4
G = < a,b,c,d,e | a12=1, b2=c2=e2=a6, d3=a4, ab=ba, ac=ca, ad=da, eae-1=a-1, cbc-1=a6b, dbd-1=a6bc, ebe-1=bc, dcd-1=b, ece-1=a6c, ede-1=a8d2 >
Subgroups: 303 in 61 conjugacy classes, 15 normal (13 characteristic)
C1, C2, C2, C3, C4, C4, C22, C6, C6, C8, C2×C4, D4, Q8, Q8, C9, Dic3, C12, C12, C2×C6, M4(2), SD16, Q16, C2×Q8, C4○D4, C18, C3⋊C8, Dic6, C2×Dic3, C2×C12, C3×D4, C3×Q8, C8.C22, Dic9, C36, C4.Dic3, D4.S3, C3⋊Q16, C2×Dic6, C3×C4○D4, Q8⋊C9, Dic18, Q8.14D6, Q8.D9, Q8.C18, C12.3S4
Quotients: C1, C2, C22, S3, D6, D9, S4, D18, C2×S4, C3.S4, C4.S4, C2×C3.S4, C12.3S4
Character table of C12.3S4
| class | 1 | 2A | 2B | 3 | 4A | 4B | 4C | 4D | 6A | 6B | 8A | 8B | 9A | 9B | 9C | 12A | 12B | 12C | 18A | 18B | 18C | 36A | 36B | 36C | 36D | 36E | 36F | |
| size | 1 | 1 | 6 | 2 | 2 | 6 | 36 | 36 | 2 | 12 | 36 | 36 | 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 | 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 | 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 | 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 | linear of order 2 |
| ρ5 | 2 | 2 | -2 | 2 | -2 | 2 | 0 | 0 | 2 | -2 | 0 | 0 | -1 | -1 | -1 | -2 | -2 | 2 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ6 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | 2 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ7 | 2 | 2 | 2 | -1 | 2 | 2 | 0 | 0 | -1 | -1 | 0 | 0 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | -1 | -1 | -1 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | ζ97+ζ92 | ζ95+ζ94 | ζ95+ζ94 | ζ98+ζ9 | ζ98+ζ9 | orthogonal lifted from D9 |
| ρ8 | 2 | 2 | -2 | -1 | -2 | 2 | 0 | 0 | -1 | 1 | 0 | 0 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | 1 | 1 | -1 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | -ζ97-ζ92 | -ζ97-ζ92 | -ζ95-ζ94 | -ζ95-ζ94 | -ζ98-ζ9 | -ζ98-ζ9 | orthogonal lifted from D18 |
| ρ9 | 2 | 2 | -2 | -1 | -2 | 2 | 0 | 0 | -1 | 1 | 0 | 0 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | 1 | 1 | -1 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | -ζ95-ζ94 | -ζ95-ζ94 | -ζ98-ζ9 | -ζ98-ζ9 | -ζ97-ζ92 | -ζ97-ζ92 | orthogonal lifted from D18 |
| ρ10 | 2 | 2 | -2 | -1 | -2 | 2 | 0 | 0 | -1 | 1 | 0 | 0 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | 1 | 1 | -1 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | -ζ98-ζ9 | -ζ98-ζ9 | -ζ97-ζ92 | -ζ97-ζ92 | -ζ95-ζ94 | -ζ95-ζ94 | orthogonal lifted from D18 |
| ρ11 | 2 | 2 | 2 | -1 | 2 | 2 | 0 | 0 | -1 | -1 | 0 | 0 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | -1 | -1 | -1 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | ζ98+ζ9 | ζ97+ζ92 | ζ97+ζ92 | ζ95+ζ94 | ζ95+ζ94 | orthogonal lifted from D9 |
| ρ12 | 2 | 2 | 2 | -1 | 2 | 2 | 0 | 0 | -1 | -1 | 0 | 0 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | -1 | -1 | -1 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ95+ζ94 | ζ98+ζ9 | ζ98+ζ9 | ζ97+ζ92 | ζ97+ζ92 | orthogonal lifted from D9 |
| ρ13 | 3 | 3 | 1 | 3 | -3 | -1 | -1 | 1 | 3 | 1 | -1 | 1 | 0 | 0 | 0 | -3 | -3 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×S4 |
| ρ14 | 3 | 3 | -1 | 3 | 3 | -1 | -1 | -1 | 3 | -1 | 1 | 1 | 0 | 0 | 0 | 3 | 3 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S4 |
| ρ15 | 3 | 3 | -1 | 3 | 3 | -1 | 1 | 1 | 3 | -1 | -1 | -1 | 0 | 0 | 0 | 3 | 3 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S4 |
| ρ16 | 3 | 3 | 1 | 3 | -3 | -1 | 1 | -1 | 3 | 1 | 1 | -1 | 0 | 0 | 0 | -3 | -3 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×S4 |
| ρ17 | 4 | -4 | 0 | 4 | 0 | 0 | 0 | 0 | -4 | 0 | 0 | 0 | -2 | -2 | -2 | 0 | 0 | 0 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from C4.S4, Schur index 2 |
| ρ18 | 4 | -4 | 0 | 4 | 0 | 0 | 0 | 0 | -4 | 0 | 0 | 0 | 1 | 1 | 1 | 0 | 0 | 0 | -1 | -1 | -1 | -√3 | √3 | -√3 | √3 | -√3 | √3 | symplectic lifted from C4.S4, Schur index 2 |
| ρ19 | 4 | -4 | 0 | 4 | 0 | 0 | 0 | 0 | -4 | 0 | 0 | 0 | 1 | 1 | 1 | 0 | 0 | 0 | -1 | -1 | -1 | √3 | -√3 | √3 | -√3 | √3 | -√3 | symplectic lifted from C4.S4, Schur index 2 |
| ρ20 | 4 | -4 | 0 | -2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | -ζ98-ζ9 | -ζ97-ζ92 | -ζ95-ζ94 | -2√3 | 2√3 | 0 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | -ζ43ζ98+ζ43ζ9 | ζ43ζ98-ζ43ζ9 | -ζ4ζ97+ζ4ζ92 | ζ4ζ97-ζ4ζ92 | ζ4ζ95-ζ4ζ94 | -ζ4ζ95+ζ4ζ94 | symplectic faithful, Schur index 2 |
| ρ21 | 4 | -4 | 0 | -2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | -ζ98-ζ9 | -ζ97-ζ92 | -ζ95-ζ94 | 2√3 | -2√3 | 0 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ43ζ98-ζ43ζ9 | -ζ43ζ98+ζ43ζ9 | ζ4ζ97-ζ4ζ92 | -ζ4ζ97+ζ4ζ92 | -ζ4ζ95+ζ4ζ94 | ζ4ζ95-ζ4ζ94 | symplectic faithful, Schur index 2 |
| ρ22 | 4 | -4 | 0 | -2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | -ζ97-ζ92 | -ζ95-ζ94 | -ζ98-ζ9 | -2√3 | 2√3 | 0 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | -ζ4ζ97+ζ4ζ92 | ζ4ζ97-ζ4ζ92 | ζ4ζ95-ζ4ζ94 | -ζ4ζ95+ζ4ζ94 | -ζ43ζ98+ζ43ζ9 | ζ43ζ98-ζ43ζ9 | symplectic faithful, Schur index 2 |
| ρ23 | 4 | -4 | 0 | -2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | -ζ97-ζ92 | -ζ95-ζ94 | -ζ98-ζ9 | 2√3 | -2√3 | 0 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | ζ4ζ97-ζ4ζ92 | -ζ4ζ97+ζ4ζ92 | -ζ4ζ95+ζ4ζ94 | ζ4ζ95-ζ4ζ94 | ζ43ζ98-ζ43ζ9 | -ζ43ζ98+ζ43ζ9 | symplectic faithful, Schur index 2 |
| ρ24 | 4 | -4 | 0 | -2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | -ζ95-ζ94 | -ζ98-ζ9 | -ζ97-ζ92 | 2√3 | -2√3 | 0 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | -ζ4ζ95+ζ4ζ94 | ζ4ζ95-ζ4ζ94 | ζ43ζ98-ζ43ζ9 | -ζ43ζ98+ζ43ζ9 | ζ4ζ97-ζ4ζ92 | -ζ4ζ97+ζ4ζ92 | symplectic faithful, Schur index 2 |
| ρ25 | 4 | -4 | 0 | -2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | -ζ95-ζ94 | -ζ98-ζ9 | -ζ97-ζ92 | -2√3 | 2√3 | 0 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | ζ4ζ95-ζ4ζ94 | -ζ4ζ95+ζ4ζ94 | -ζ43ζ98+ζ43ζ9 | ζ43ζ98-ζ43ζ9 | -ζ4ζ97+ζ4ζ92 | ζ4ζ97-ζ4ζ92 | symplectic faithful, Schur index 2 |
| ρ26 | 6 | 6 | -2 | -3 | 6 | -2 | 0 | 0 | -3 | 1 | 0 | 0 | 0 | 0 | 0 | -3 | -3 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C3.S4 |
| ρ27 | 6 | 6 | 2 | -3 | -6 | -2 | 0 | 0 | -3 | -1 | 0 | 0 | 0 | 0 | 0 | 3 | 3 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×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 10 7 4)(2 11 8 5)(3 12 9 6)(13 81 19 75)(14 82 20 76)(15 83 21 77)(16 84 22 78)(17 73 23 79)(18 74 24 80)(25 91 31 85)(26 92 32 86)(27 93 33 87)(28 94 34 88)(29 95 35 89)(30 96 36 90)(37 46 43 40)(38 47 44 41)(39 48 45 42)(49 122 55 128)(50 123 56 129)(51 124 57 130)(52 125 58 131)(53 126 59 132)(54 127 60 121)(61 64 67 70)(62 65 68 71)(63 66 69 72)(97 119 103 113)(98 120 104 114)(99 109 105 115)(100 110 106 116)(101 111 107 117)(102 112 108 118)(133 136 139 142)(134 137 140 143)(135 138 141 144)
(1 137 7 143)(2 138 8 144)(3 139 9 133)(4 140 10 134)(5 141 11 135)(6 142 12 136)(13 78 19 84)(14 79 20 73)(15 80 21 74)(16 81 22 75)(17 82 23 76)(18 83 24 77)(25 28 31 34)(26 29 32 35)(27 30 33 36)(37 70 43 64)(38 71 44 65)(39 72 45 66)(40 61 46 67)(41 62 47 68)(42 63 48 69)(49 131 55 125)(50 132 56 126)(51 121 57 127)(52 122 58 128)(53 123 59 129)(54 124 60 130)(85 94 91 88)(86 95 92 89)(87 96 93 90)(97 100 103 106)(98 101 104 107)(99 102 105 108)(109 118 115 112)(110 119 116 113)(111 120 117 114)
(1 86 131 5 90 123 9 94 127)(2 87 132 6 91 124 10 95 128)(3 88 121 7 92 125 11 96 129)(4 89 122 8 93 126 12 85 130)(13 62 105 17 66 97 21 70 101)(14 63 106 18 67 98 22 71 102)(15 64 107 19 68 99 23 72 103)(16 65 108 20 69 100 24 61 104)(25 54 143 29 58 135 33 50 139)(26 55 144 30 59 136 34 51 140)(27 56 133 31 60 137 35 52 141)(28 57 134 32 49 138 36 53 142)(37 120 75 41 112 79 45 116 83)(38 109 76 42 113 80 46 117 84)(39 110 77 43 114 81 47 118 73)(40 111 78 44 115 82 48 119 74)
(1 73 7 79)(2 84 8 78)(3 83 9 77)(4 82 10 76)(5 81 11 75)(6 80 12 74)(13 138 19 144)(14 137 20 143)(15 136 21 142)(16 135 22 141)(17 134 23 140)(18 133 24 139)(25 106 31 100)(26 105 32 99)(27 104 33 98)(28 103 34 97)(29 102 35 108)(30 101 36 107)(37 123 43 129)(38 122 44 128)(39 121 45 127)(40 132 46 126)(41 131 47 125)(42 130 48 124)(49 68 55 62)(50 67 56 61)(51 66 57 72)(52 65 58 71)(53 64 59 70)(54 63 60 69)(85 119 91 113)(86 118 92 112)(87 117 93 111)(88 116 94 110)(89 115 95 109)(90 114 96 120)
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), (1,10,7,4)(2,11,8,5)(3,12,9,6)(13,81,19,75)(14,82,20,76)(15,83,21,77)(16,84,22,78)(17,73,23,79)(18,74,24,80)(25,91,31,85)(26,92,32,86)(27,93,33,87)(28,94,34,88)(29,95,35,89)(30,96,36,90)(37,46,43,40)(38,47,44,41)(39,48,45,42)(49,122,55,128)(50,123,56,129)(51,124,57,130)(52,125,58,131)(53,126,59,132)(54,127,60,121)(61,64,67,70)(62,65,68,71)(63,66,69,72)(97,119,103,113)(98,120,104,114)(99,109,105,115)(100,110,106,116)(101,111,107,117)(102,112,108,118)(133,136,139,142)(134,137,140,143)(135,138,141,144), (1,137,7,143)(2,138,8,144)(3,139,9,133)(4,140,10,134)(5,141,11,135)(6,142,12,136)(13,78,19,84)(14,79,20,73)(15,80,21,74)(16,81,22,75)(17,82,23,76)(18,83,24,77)(25,28,31,34)(26,29,32,35)(27,30,33,36)(37,70,43,64)(38,71,44,65)(39,72,45,66)(40,61,46,67)(41,62,47,68)(42,63,48,69)(49,131,55,125)(50,132,56,126)(51,121,57,127)(52,122,58,128)(53,123,59,129)(54,124,60,130)(85,94,91,88)(86,95,92,89)(87,96,93,90)(97,100,103,106)(98,101,104,107)(99,102,105,108)(109,118,115,112)(110,119,116,113)(111,120,117,114), (1,86,131,5,90,123,9,94,127)(2,87,132,6,91,124,10,95,128)(3,88,121,7,92,125,11,96,129)(4,89,122,8,93,126,12,85,130)(13,62,105,17,66,97,21,70,101)(14,63,106,18,67,98,22,71,102)(15,64,107,19,68,99,23,72,103)(16,65,108,20,69,100,24,61,104)(25,54,143,29,58,135,33,50,139)(26,55,144,30,59,136,34,51,140)(27,56,133,31,60,137,35,52,141)(28,57,134,32,49,138,36,53,142)(37,120,75,41,112,79,45,116,83)(38,109,76,42,113,80,46,117,84)(39,110,77,43,114,81,47,118,73)(40,111,78,44,115,82,48,119,74), (1,73,7,79)(2,84,8,78)(3,83,9,77)(4,82,10,76)(5,81,11,75)(6,80,12,74)(13,138,19,144)(14,137,20,143)(15,136,21,142)(16,135,22,141)(17,134,23,140)(18,133,24,139)(25,106,31,100)(26,105,32,99)(27,104,33,98)(28,103,34,97)(29,102,35,108)(30,101,36,107)(37,123,43,129)(38,122,44,128)(39,121,45,127)(40,132,46,126)(41,131,47,125)(42,130,48,124)(49,68,55,62)(50,67,56,61)(51,66,57,72)(52,65,58,71)(53,64,59,70)(54,63,60,69)(85,119,91,113)(86,118,92,112)(87,117,93,111)(88,116,94,110)(89,115,95,109)(90,114,96,120)>;
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), (1,10,7,4)(2,11,8,5)(3,12,9,6)(13,81,19,75)(14,82,20,76)(15,83,21,77)(16,84,22,78)(17,73,23,79)(18,74,24,80)(25,91,31,85)(26,92,32,86)(27,93,33,87)(28,94,34,88)(29,95,35,89)(30,96,36,90)(37,46,43,40)(38,47,44,41)(39,48,45,42)(49,122,55,128)(50,123,56,129)(51,124,57,130)(52,125,58,131)(53,126,59,132)(54,127,60,121)(61,64,67,70)(62,65,68,71)(63,66,69,72)(97,119,103,113)(98,120,104,114)(99,109,105,115)(100,110,106,116)(101,111,107,117)(102,112,108,118)(133,136,139,142)(134,137,140,143)(135,138,141,144), (1,137,7,143)(2,138,8,144)(3,139,9,133)(4,140,10,134)(5,141,11,135)(6,142,12,136)(13,78,19,84)(14,79,20,73)(15,80,21,74)(16,81,22,75)(17,82,23,76)(18,83,24,77)(25,28,31,34)(26,29,32,35)(27,30,33,36)(37,70,43,64)(38,71,44,65)(39,72,45,66)(40,61,46,67)(41,62,47,68)(42,63,48,69)(49,131,55,125)(50,132,56,126)(51,121,57,127)(52,122,58,128)(53,123,59,129)(54,124,60,130)(85,94,91,88)(86,95,92,89)(87,96,93,90)(97,100,103,106)(98,101,104,107)(99,102,105,108)(109,118,115,112)(110,119,116,113)(111,120,117,114), (1,86,131,5,90,123,9,94,127)(2,87,132,6,91,124,10,95,128)(3,88,121,7,92,125,11,96,129)(4,89,122,8,93,126,12,85,130)(13,62,105,17,66,97,21,70,101)(14,63,106,18,67,98,22,71,102)(15,64,107,19,68,99,23,72,103)(16,65,108,20,69,100,24,61,104)(25,54,143,29,58,135,33,50,139)(26,55,144,30,59,136,34,51,140)(27,56,133,31,60,137,35,52,141)(28,57,134,32,49,138,36,53,142)(37,120,75,41,112,79,45,116,83)(38,109,76,42,113,80,46,117,84)(39,110,77,43,114,81,47,118,73)(40,111,78,44,115,82,48,119,74), (1,73,7,79)(2,84,8,78)(3,83,9,77)(4,82,10,76)(5,81,11,75)(6,80,12,74)(13,138,19,144)(14,137,20,143)(15,136,21,142)(16,135,22,141)(17,134,23,140)(18,133,24,139)(25,106,31,100)(26,105,32,99)(27,104,33,98)(28,103,34,97)(29,102,35,108)(30,101,36,107)(37,123,43,129)(38,122,44,128)(39,121,45,127)(40,132,46,126)(41,131,47,125)(42,130,48,124)(49,68,55,62)(50,67,56,61)(51,66,57,72)(52,65,58,71)(53,64,59,70)(54,63,60,69)(85,119,91,113)(86,118,92,112)(87,117,93,111)(88,116,94,110)(89,115,95,109)(90,114,96,120) );
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)], [(1,10,7,4),(2,11,8,5),(3,12,9,6),(13,81,19,75),(14,82,20,76),(15,83,21,77),(16,84,22,78),(17,73,23,79),(18,74,24,80),(25,91,31,85),(26,92,32,86),(27,93,33,87),(28,94,34,88),(29,95,35,89),(30,96,36,90),(37,46,43,40),(38,47,44,41),(39,48,45,42),(49,122,55,128),(50,123,56,129),(51,124,57,130),(52,125,58,131),(53,126,59,132),(54,127,60,121),(61,64,67,70),(62,65,68,71),(63,66,69,72),(97,119,103,113),(98,120,104,114),(99,109,105,115),(100,110,106,116),(101,111,107,117),(102,112,108,118),(133,136,139,142),(134,137,140,143),(135,138,141,144)], [(1,137,7,143),(2,138,8,144),(3,139,9,133),(4,140,10,134),(5,141,11,135),(6,142,12,136),(13,78,19,84),(14,79,20,73),(15,80,21,74),(16,81,22,75),(17,82,23,76),(18,83,24,77),(25,28,31,34),(26,29,32,35),(27,30,33,36),(37,70,43,64),(38,71,44,65),(39,72,45,66),(40,61,46,67),(41,62,47,68),(42,63,48,69),(49,131,55,125),(50,132,56,126),(51,121,57,127),(52,122,58,128),(53,123,59,129),(54,124,60,130),(85,94,91,88),(86,95,92,89),(87,96,93,90),(97,100,103,106),(98,101,104,107),(99,102,105,108),(109,118,115,112),(110,119,116,113),(111,120,117,114)], [(1,86,131,5,90,123,9,94,127),(2,87,132,6,91,124,10,95,128),(3,88,121,7,92,125,11,96,129),(4,89,122,8,93,126,12,85,130),(13,62,105,17,66,97,21,70,101),(14,63,106,18,67,98,22,71,102),(15,64,107,19,68,99,23,72,103),(16,65,108,20,69,100,24,61,104),(25,54,143,29,58,135,33,50,139),(26,55,144,30,59,136,34,51,140),(27,56,133,31,60,137,35,52,141),(28,57,134,32,49,138,36,53,142),(37,120,75,41,112,79,45,116,83),(38,109,76,42,113,80,46,117,84),(39,110,77,43,114,81,47,118,73),(40,111,78,44,115,82,48,119,74)], [(1,73,7,79),(2,84,8,78),(3,83,9,77),(4,82,10,76),(5,81,11,75),(6,80,12,74),(13,138,19,144),(14,137,20,143),(15,136,21,142),(16,135,22,141),(17,134,23,140),(18,133,24,139),(25,106,31,100),(26,105,32,99),(27,104,33,98),(28,103,34,97),(29,102,35,108),(30,101,36,107),(37,123,43,129),(38,122,44,128),(39,121,45,127),(40,132,46,126),(41,131,47,125),(42,130,48,124),(49,68,55,62),(50,67,56,61),(51,66,57,72),(52,65,58,71),(53,64,59,70),(54,63,60,69),(85,119,91,113),(86,118,92,112),(87,117,93,111),(88,116,94,110),(89,115,95,109),(90,114,96,120)]])
Matrix representation of C12.3S4 ►in GL4(𝔽73) generated by
| 70 | 0 | 0 | 0 |
| 0 | 70 | 0 | 0 |
| 14 | 71 | 24 | 0 |
| 49 | 43 | 0 | 24 |
| 27 | 0 | 0 | 0 |
| 27 | 46 | 0 | 0 |
| 52 | 5 | 0 | 1 |
| 67 | 62 | 72 | 0 |
| 27 | 19 | 0 | 0 |
| 0 | 46 | 0 | 0 |
| 10 | 56 | 0 | 46 |
| 10 | 70 | 46 | 0 |
| 19 | 52 | 0 | 0 |
| 47 | 52 | 0 | 0 |
| 48 | 53 | 43 | 43 |
| 33 | 68 | 7 | 66 |
| 26 | 38 | 71 | 0 |
| 4 | 26 | 72 | 1 |
| 13 | 39 | 28 | 19 |
| 24 | 13 | 62 | 66 |
G:=sub<GL(4,GF(73))| [70,0,14,49,0,70,71,43,0,0,24,0,0,0,0,24],[27,27,52,67,0,46,5,62,0,0,0,72,0,0,1,0],[27,0,10,10,19,46,56,70,0,0,0,46,0,0,46,0],[19,47,48,33,52,52,53,68,0,0,43,7,0,0,43,66],[26,4,13,24,38,26,39,13,71,72,28,62,0,1,19,66] >;
C12.3S4 in GAP, Magma, Sage, TeX
C_{12}._3S_4 % in TeX
G:=Group("C12.3S4"); // GroupNames label
G:=SmallGroup(288,338);
// by ID
G=gap.SmallGroup(288,338);
# by ID
G:=PCGroup([7,-2,-2,-3,-3,-2,2,-2,1008,2045,1016,422,142,675,2524,1908,172,1517,1153,285,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^12=1,b^2=c^2=e^2=a^6,d^3=a^4,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e^-1=a^-1,c*b*c^-1=a^6*b,d*b*d^-1=a^6*b*c,e*b*e^-1=b*c,d*c*d^-1=b,e*c*e^-1=a^6*c,e*d*e^-1=a^8*d^2>;
// generators/relations
Export