metabelian, supersoluble, monomial
Aliases: C24.28D6, (C3×Q16)⋊3S3, C32⋊5D8⋊9C2, Q16⋊3(C3⋊S3), C6.127(S3×D4), (C3×Q8).40D6, C32⋊23(C4○D8), C3⋊Dic3.71D4, (C32×Q16)⋊6C2, C3⋊4(D24⋊C2), C32⋊11SD16⋊8C2, C12.26D6⋊5C2, (C3×C24).31C22, C12.96(C22×S3), (C3×C12).100C23, C12⋊S3.20C22, C32⋊4C8.29C22, (Q8×C32).20C22, (C8×C3⋊S3)⋊5C2, C8.10(C2×C3⋊S3), C2.24(D4×C3⋊S3), (C2×C3⋊S3).48D4, Q8.10(C2×C3⋊S3), (C3×C6).248(C2×D4), C4.10(C22×C3⋊S3), (C4×C3⋊S3).75C22, SmallGroup(288,776)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24.28D6
G = < a,b,c | a24=c2=1, b6=a12, bab-1=a7, cac=a-1, cbc=a12b5 >
Subgroups: 836 in 186 conjugacy classes, 55 normal (17 characteristic)
C1, C2, C2, C3, C4, C4, C22, S3, C6, C8, C8, C2×C4, D4, Q8, C32, Dic3, C12, C12, D6, C2×C8, D8, SD16, Q16, C4○D4, C3⋊S3, C3×C6, C3⋊C8, C24, C4×S3, D12, C3×Q8, C4○D8, C3⋊Dic3, C3×C12, C3×C12, C2×C3⋊S3, C2×C3⋊S3, S3×C8, D24, Q8⋊2S3, C3×Q16, Q8⋊3S3, C32⋊4C8, C3×C24, C4×C3⋊S3, C4×C3⋊S3, C12⋊S3, C12⋊S3, Q8×C32, D24⋊C2, C8×C3⋊S3, C32⋊5D8, C32⋊11SD16, C32×Q16, C12.26D6, C24.28D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C3⋊S3, C22×S3, C4○D8, C2×C3⋊S3, S3×D4, C22×C3⋊S3, D24⋊C2, D4×C3⋊S3, C24.28D6
►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)
(1 141 38 70 86 101 13 129 26 58 74 113)(2 124 39 53 87 108 14 136 27 65 75 120)(3 131 40 60 88 115 15 143 28 72 76 103)(4 138 41 67 89 98 16 126 29 55 77 110)(5 121 42 50 90 105 17 133 30 62 78 117)(6 128 43 57 91 112 18 140 31 69 79 100)(7 135 44 64 92 119 19 123 32 52 80 107)(8 142 45 71 93 102 20 130 33 59 81 114)(9 125 46 54 94 109 21 137 34 66 82 97)(10 132 47 61 95 116 22 144 35 49 83 104)(11 139 48 68 96 99 23 127 36 56 84 111)(12 122 25 51 73 106 24 134 37 63 85 118)
(1 29)(2 28)(3 27)(4 26)(5 25)(6 48)(7 47)(8 46)(9 45)(10 44)(11 43)(12 42)(13 41)(14 40)(15 39)(16 38)(17 37)(18 36)(19 35)(20 34)(21 33)(22 32)(23 31)(24 30)(49 119)(50 118)(51 117)(52 116)(53 115)(54 114)(55 113)(56 112)(57 111)(58 110)(59 109)(60 108)(61 107)(62 106)(63 105)(64 104)(65 103)(66 102)(67 101)(68 100)(69 99)(70 98)(71 97)(72 120)(73 78)(74 77)(75 76)(79 96)(80 95)(81 94)(82 93)(83 92)(84 91)(85 90)(86 89)(87 88)(121 122)(123 144)(124 143)(125 142)(126 141)(127 140)(128 139)(129 138)(130 137)(131 136)(132 135)(133 134)
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), (1,141,38,70,86,101,13,129,26,58,74,113)(2,124,39,53,87,108,14,136,27,65,75,120)(3,131,40,60,88,115,15,143,28,72,76,103)(4,138,41,67,89,98,16,126,29,55,77,110)(5,121,42,50,90,105,17,133,30,62,78,117)(6,128,43,57,91,112,18,140,31,69,79,100)(7,135,44,64,92,119,19,123,32,52,80,107)(8,142,45,71,93,102,20,130,33,59,81,114)(9,125,46,54,94,109,21,137,34,66,82,97)(10,132,47,61,95,116,22,144,35,49,83,104)(11,139,48,68,96,99,23,127,36,56,84,111)(12,122,25,51,73,106,24,134,37,63,85,118), (1,29)(2,28)(3,27)(4,26)(5,25)(6,48)(7,47)(8,46)(9,45)(10,44)(11,43)(12,42)(13,41)(14,40)(15,39)(16,38)(17,37)(18,36)(19,35)(20,34)(21,33)(22,32)(23,31)(24,30)(49,119)(50,118)(51,117)(52,116)(53,115)(54,114)(55,113)(56,112)(57,111)(58,110)(59,109)(60,108)(61,107)(62,106)(63,105)(64,104)(65,103)(66,102)(67,101)(68,100)(69,99)(70,98)(71,97)(72,120)(73,78)(74,77)(75,76)(79,96)(80,95)(81,94)(82,93)(83,92)(84,91)(85,90)(86,89)(87,88)(121,122)(123,144)(124,143)(125,142)(126,141)(127,140)(128,139)(129,138)(130,137)(131,136)(132,135)(133,134)>;
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), (1,141,38,70,86,101,13,129,26,58,74,113)(2,124,39,53,87,108,14,136,27,65,75,120)(3,131,40,60,88,115,15,143,28,72,76,103)(4,138,41,67,89,98,16,126,29,55,77,110)(5,121,42,50,90,105,17,133,30,62,78,117)(6,128,43,57,91,112,18,140,31,69,79,100)(7,135,44,64,92,119,19,123,32,52,80,107)(8,142,45,71,93,102,20,130,33,59,81,114)(9,125,46,54,94,109,21,137,34,66,82,97)(10,132,47,61,95,116,22,144,35,49,83,104)(11,139,48,68,96,99,23,127,36,56,84,111)(12,122,25,51,73,106,24,134,37,63,85,118), (1,29)(2,28)(3,27)(4,26)(5,25)(6,48)(7,47)(8,46)(9,45)(10,44)(11,43)(12,42)(13,41)(14,40)(15,39)(16,38)(17,37)(18,36)(19,35)(20,34)(21,33)(22,32)(23,31)(24,30)(49,119)(50,118)(51,117)(52,116)(53,115)(54,114)(55,113)(56,112)(57,111)(58,110)(59,109)(60,108)(61,107)(62,106)(63,105)(64,104)(65,103)(66,102)(67,101)(68,100)(69,99)(70,98)(71,97)(72,120)(73,78)(74,77)(75,76)(79,96)(80,95)(81,94)(82,93)(83,92)(84,91)(85,90)(86,89)(87,88)(121,122)(123,144)(124,143)(125,142)(126,141)(127,140)(128,139)(129,138)(130,137)(131,136)(132,135)(133,134) );
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)], [(1,141,38,70,86,101,13,129,26,58,74,113),(2,124,39,53,87,108,14,136,27,65,75,120),(3,131,40,60,88,115,15,143,28,72,76,103),(4,138,41,67,89,98,16,126,29,55,77,110),(5,121,42,50,90,105,17,133,30,62,78,117),(6,128,43,57,91,112,18,140,31,69,79,100),(7,135,44,64,92,119,19,123,32,52,80,107),(8,142,45,71,93,102,20,130,33,59,81,114),(9,125,46,54,94,109,21,137,34,66,82,97),(10,132,47,61,95,116,22,144,35,49,83,104),(11,139,48,68,96,99,23,127,36,56,84,111),(12,122,25,51,73,106,24,134,37,63,85,118)], [(1,29),(2,28),(3,27),(4,26),(5,25),(6,48),(7,47),(8,46),(9,45),(10,44),(11,43),(12,42),(13,41),(14,40),(15,39),(16,38),(17,37),(18,36),(19,35),(20,34),(21,33),(22,32),(23,31),(24,30),(49,119),(50,118),(51,117),(52,116),(53,115),(54,114),(55,113),(56,112),(57,111),(58,110),(59,109),(60,108),(61,107),(62,106),(63,105),(64,104),(65,103),(66,102),(67,101),(68,100),(69,99),(70,98),(71,97),(72,120),(73,78),(74,77),(75,76),(79,96),(80,95),(81,94),(82,93),(83,92),(84,91),(85,90),(86,89),(87,88),(121,122),(123,144),(124,143),(125,142),(126,141),(127,140),(128,139),(129,138),(130,137),(131,136),(132,135),(133,134)]])
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3A | 3B | 3C | 3D | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 6C | 6D | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | ··· | 12L | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 18 | 36 | 36 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 9 | 9 | 2 | 2 | 2 | 2 | 2 | 2 | 18 | 18 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | C4○D8 | S3×D4 | D24⋊C2 |
| kernel | C24.28D6 | C8×C3⋊S3 | C32⋊5D8 | C32⋊11SD16 | C32×Q16 | C12.26D6 | C3×Q16 | C3⋊Dic3 | C2×C3⋊S3 | C24 | C3×Q8 | C32 | C6 | C3 |
| # reps | 1 | 1 | 1 | 2 | 1 | 2 | 4 | 1 | 1 | 4 | 8 | 4 | 4 | 8 |
Matrix representation of C24.28D6 ►in GL6(𝔽73)
| 0 | 1 | 0 | 0 | 0 | 0 |
| 72 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 57 |
| 0 | 0 | 0 | 0 | 16 | 16 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 1 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 6 | 6 |
| 0 | 0 | 0 | 0 | 6 | 67 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 57 | 16 |
| 0 | 0 | 0 | 0 | 16 | 16 |
G:=sub<GL(6,GF(73))| [0,72,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,16,0,0,0,0,57,16],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,72,72,0,0,0,0,0,0,6,6,0,0,0,0,6,67],[0,72,0,0,0,0,72,0,0,0,0,0,0,0,1,1,0,0,0,0,0,72,0,0,0,0,0,0,57,16,0,0,0,0,16,16] >;
C24.28D6 in GAP, Magma, Sage, TeX
C_{24}._{28}D_6 % in TeX
G:=Group("C24.28D6"); // GroupNames label
G:=SmallGroup(288,776);
// by ID
G=gap.SmallGroup(288,776);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,120,422,135,100,346,185,80,2693,9414]);
// Polycyclic
G:=Group<a,b,c|a^24=c^2=1,b^6=a^12,b*a*b^-1=a^7,c*a*c=a^-1,c*b*c=a^12*b^5>;
// generators/relations