metabelian, supersoluble, monomial
Aliases: C24.47D6, C12.40(C4×S3), C24⋊S3⋊10C2, C12⋊S3.3C4, C3⋊4(D12.C4), (C2×C12).146D6, C32⋊12(C8○D4), (C3×M4(2))⋊6S3, C62.64(C2×C4), C32⋊7D4.3C4, M4(2)⋊5(C3⋊S3), C32⋊4Q8.3C4, (C3×C24).49C22, (C3×C12).179C23, (C6×C12).137C22, C12.210(C22×S3), C12.59D6.6C2, (C32×M4(2))⋊10C2, C32⋊4C8.40C22, C4.5(C4×C3⋊S3), C6.76(S3×C2×C4), (C8×C3⋊S3)⋊16C2, C8.12(C2×C3⋊S3), (C2×C6).23(C4×S3), C22.1(C4×C3⋊S3), (C3×C12).74(C2×C4), (C2×C32⋊4C8)⋊8C2, C4.39(C22×C3⋊S3), (C4×C3⋊S3).94C22, C3⋊Dic3.39(C2×C4), (C3×C6).107(C22×C4), C2.17(C2×C4×C3⋊S3), (C2×C4).45(C2×C3⋊S3), (C2×C3⋊S3).33(C2×C4), SmallGroup(288,764)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24.47D6
G = < a,b,c | a24=b6=1, c2=a12, bab-1=a13, cac-1=a17, cbc-1=a12b-1 >
Subgroups: 572 in 186 conjugacy classes, 73 normal (21 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C8, C2×C4, C2×C4, D4, Q8, C32, Dic3, C12, D6, C2×C6, C2×C8, M4(2), M4(2), C4○D4, C3⋊S3, C3×C6, C3×C6, C3⋊C8, C24, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C8○D4, C3⋊Dic3, C3×C12, C2×C3⋊S3, C62, S3×C8, C8⋊S3, C2×C3⋊C8, C3×M4(2), C4○D12, C32⋊4C8, C3×C24, C32⋊4Q8, C4×C3⋊S3, C12⋊S3, C32⋊7D4, C6×C12, D12.C4, C8×C3⋊S3, C24⋊S3, C2×C32⋊4C8, C32×M4(2), C12.59D6, C24.47D6
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D6, C22×C4, C3⋊S3, C4×S3, C22×S3, C8○D4, C2×C3⋊S3, S3×C2×C4, C4×C3⋊S3, C22×C3⋊S3, D12.C4, C2×C4×C3⋊S3, C24.47D6
►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 81 39 54 134 97)(2 94 40 67 135 110)(3 83 41 56 136 99)(4 96 42 69 137 112)(5 85 43 58 138 101)(6 74 44 71 139 114)(7 87 45 60 140 103)(8 76 46 49 141 116)(9 89 47 62 142 105)(10 78 48 51 143 118)(11 91 25 64 144 107)(12 80 26 53 121 120)(13 93 27 66 122 109)(14 82 28 55 123 98)(15 95 29 68 124 111)(16 84 30 57 125 100)(17 73 31 70 126 113)(18 86 32 59 127 102)(19 75 33 72 128 115)(20 88 34 61 129 104)(21 77 35 50 130 117)(22 90 36 63 131 106)(23 79 37 52 132 119)(24 92 38 65 133 108)
(1 33 13 45)(2 26 14 38)(3 43 15 31)(4 36 16 48)(5 29 17 41)(6 46 18 34)(7 39 19 27)(8 32 20 44)(9 25 21 37)(10 42 22 30)(11 35 23 47)(12 28 24 40)(49 114 61 102)(50 107 62 119)(51 100 63 112)(52 117 64 105)(53 110 65 98)(54 103 66 115)(55 120 67 108)(56 113 68 101)(57 106 69 118)(58 99 70 111)(59 116 71 104)(60 109 72 97)(73 95 85 83)(74 88 86 76)(75 81 87 93)(77 91 89 79)(78 84 90 96)(80 94 92 82)(121 123 133 135)(122 140 134 128)(124 126 136 138)(125 143 137 131)(127 129 139 141)(130 132 142 144)
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), (1,81,39,54,134,97)(2,94,40,67,135,110)(3,83,41,56,136,99)(4,96,42,69,137,112)(5,85,43,58,138,101)(6,74,44,71,139,114)(7,87,45,60,140,103)(8,76,46,49,141,116)(9,89,47,62,142,105)(10,78,48,51,143,118)(11,91,25,64,144,107)(12,80,26,53,121,120)(13,93,27,66,122,109)(14,82,28,55,123,98)(15,95,29,68,124,111)(16,84,30,57,125,100)(17,73,31,70,126,113)(18,86,32,59,127,102)(19,75,33,72,128,115)(20,88,34,61,129,104)(21,77,35,50,130,117)(22,90,36,63,131,106)(23,79,37,52,132,119)(24,92,38,65,133,108), (1,33,13,45)(2,26,14,38)(3,43,15,31)(4,36,16,48)(5,29,17,41)(6,46,18,34)(7,39,19,27)(8,32,20,44)(9,25,21,37)(10,42,22,30)(11,35,23,47)(12,28,24,40)(49,114,61,102)(50,107,62,119)(51,100,63,112)(52,117,64,105)(53,110,65,98)(54,103,66,115)(55,120,67,108)(56,113,68,101)(57,106,69,118)(58,99,70,111)(59,116,71,104)(60,109,72,97)(73,95,85,83)(74,88,86,76)(75,81,87,93)(77,91,89,79)(78,84,90,96)(80,94,92,82)(121,123,133,135)(122,140,134,128)(124,126,136,138)(125,143,137,131)(127,129,139,141)(130,132,142,144)>;
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), (1,81,39,54,134,97)(2,94,40,67,135,110)(3,83,41,56,136,99)(4,96,42,69,137,112)(5,85,43,58,138,101)(6,74,44,71,139,114)(7,87,45,60,140,103)(8,76,46,49,141,116)(9,89,47,62,142,105)(10,78,48,51,143,118)(11,91,25,64,144,107)(12,80,26,53,121,120)(13,93,27,66,122,109)(14,82,28,55,123,98)(15,95,29,68,124,111)(16,84,30,57,125,100)(17,73,31,70,126,113)(18,86,32,59,127,102)(19,75,33,72,128,115)(20,88,34,61,129,104)(21,77,35,50,130,117)(22,90,36,63,131,106)(23,79,37,52,132,119)(24,92,38,65,133,108), (1,33,13,45)(2,26,14,38)(3,43,15,31)(4,36,16,48)(5,29,17,41)(6,46,18,34)(7,39,19,27)(8,32,20,44)(9,25,21,37)(10,42,22,30)(11,35,23,47)(12,28,24,40)(49,114,61,102)(50,107,62,119)(51,100,63,112)(52,117,64,105)(53,110,65,98)(54,103,66,115)(55,120,67,108)(56,113,68,101)(57,106,69,118)(58,99,70,111)(59,116,71,104)(60,109,72,97)(73,95,85,83)(74,88,86,76)(75,81,87,93)(77,91,89,79)(78,84,90,96)(80,94,92,82)(121,123,133,135)(122,140,134,128)(124,126,136,138)(125,143,137,131)(127,129,139,141)(130,132,142,144) );
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)], [(1,81,39,54,134,97),(2,94,40,67,135,110),(3,83,41,56,136,99),(4,96,42,69,137,112),(5,85,43,58,138,101),(6,74,44,71,139,114),(7,87,45,60,140,103),(8,76,46,49,141,116),(9,89,47,62,142,105),(10,78,48,51,143,118),(11,91,25,64,144,107),(12,80,26,53,121,120),(13,93,27,66,122,109),(14,82,28,55,123,98),(15,95,29,68,124,111),(16,84,30,57,125,100),(17,73,31,70,126,113),(18,86,32,59,127,102),(19,75,33,72,128,115),(20,88,34,61,129,104),(21,77,35,50,130,117),(22,90,36,63,131,106),(23,79,37,52,132,119),(24,92,38,65,133,108)], [(1,33,13,45),(2,26,14,38),(3,43,15,31),(4,36,16,48),(5,29,17,41),(6,46,18,34),(7,39,19,27),(8,32,20,44),(9,25,21,37),(10,42,22,30),(11,35,23,47),(12,28,24,40),(49,114,61,102),(50,107,62,119),(51,100,63,112),(52,117,64,105),(53,110,65,98),(54,103,66,115),(55,120,67,108),(56,113,68,101),(57,106,69,118),(58,99,70,111),(59,116,71,104),(60,109,72,97),(73,95,85,83),(74,88,86,76),(75,81,87,93),(77,91,89,79),(78,84,90,96),(80,94,92,82),(121,123,133,135),(122,140,134,128),(124,126,136,138),(125,143,137,131),(127,129,139,141),(130,132,142,144)]])
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3A | 3B | 3C | 3D | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | 8J | 12A | ··· | 12H | 12I | 12J | 12K | 12L | 24A | ··· | 24P |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | ··· | 12 | 12 | 12 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 2 | 18 | 18 | 2 | 2 | 2 | 2 | 1 | 1 | 2 | 18 | 18 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 9 | 9 | 9 | 9 | 18 | 18 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | S3 | D6 | D6 | C4×S3 | C4×S3 | C8○D4 | D12.C4 |
| kernel | C24.47D6 | C8×C3⋊S3 | C24⋊S3 | C2×C32⋊4C8 | C32×M4(2) | C12.59D6 | C32⋊4Q8 | C12⋊S3 | C32⋊7D4 | C3×M4(2) | C24 | C2×C12 | C12 | C2×C6 | C32 | C3 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 8 | 4 | 8 | 8 | 4 | 8 |
Matrix representation of C24.47D6 ►in GL6(𝔽73)
| 72 | 72 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 46 | 46 | 0 | 0 |
| 0 | 0 | 27 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 63 | 0 |
| 0 | 0 | 0 | 0 | 43 | 10 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 51 | 39 |
| 0 | 0 | 0 | 0 | 55 | 22 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 72 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 27 | 0 |
| 0 | 0 | 0 | 0 | 8 | 46 |
G:=sub<GL(6,GF(73))| [72,1,0,0,0,0,72,0,0,0,0,0,0,0,46,27,0,0,0,0,46,0,0,0,0,0,0,0,63,43,0,0,0,0,0,10],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,72,1,0,0,0,0,0,0,51,55,0,0,0,0,39,22],[1,72,0,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,72,0,0,0,0,0,0,0,27,8,0,0,0,0,0,46] >;
C24.47D6 in GAP, Magma, Sage, TeX
C_{24}._{47}D_6 % in TeX
G:=Group("C24.47D6"); // GroupNames label
G:=SmallGroup(288,764);
// by ID
G=gap.SmallGroup(288,764);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,219,58,80,2693,9414]);
// Polycyclic
G:=Group<a,b,c|a^24=b^6=1,c^2=a^12,b*a*b^-1=a^13,c*a*c^-1=a^17,c*b*c^-1=a^12*b^-1>;
// generators/relations