metabelian, supersoluble, monomial
Aliases: C24.78D6, C12.61D12, C62.92D4, (C6×C24)⋊8C2, (C2×C24)⋊6S3, C3⋊4(C4○D24), (C2×C6).40D12, C6.58(C2×D12), C24⋊2S3⋊11C2, C32⋊5D8⋊11C2, (C3×C12).141D4, (C2×C12).399D6, C32⋊17(C4○D8), C32⋊5Q16⋊11C2, C12.59D6⋊1C2, (C3×C24).59C22, C4.20(C12⋊S3), C12.191(C22×S3), (C6×C12).309C22, (C3×C12).153C23, C12⋊S3.23C22, C22.1(C12⋊S3), C32⋊4Q8.23C22, (C2×C8)⋊4(C3⋊S3), C8.16(C2×C3⋊S3), (C3×C6).198(C2×D4), C4.28(C22×C3⋊S3), C2.13(C2×C12⋊S3), (C2×C4).81(C2×C3⋊S3), SmallGroup(288,761)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24.78D6
G = < a,b,c | a24=1, b6=c2=a12, ab=ba, cac-1=a11, cbc-1=b5 >
Subgroups: 836 in 186 conjugacy classes, 65 normal (21 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, Q8, C32, Dic3, C12, D6, C2×C6, C2×C8, D8, SD16, Q16, C4○D4, C3⋊S3, C3×C6, C3×C6, C24, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C4○D8, C3⋊Dic3, C3×C12, C2×C3⋊S3, C62, C24⋊C2, D24, Dic12, C2×C24, C4○D12, C3×C24, C32⋊4Q8, C4×C3⋊S3, C12⋊S3, C32⋊7D4, C6×C12, C4○D24, C24⋊2S3, C32⋊5D8, C32⋊5Q16, C6×C24, C12.59D6, C24.78D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C3⋊S3, D12, C22×S3, C4○D8, C2×C3⋊S3, C2×D12, C12⋊S3, C22×C3⋊S3, C4○D24, C2×C12⋊S3, C24.78D6
►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 92 124 56 28 100 13 80 136 68 40 112)(2 93 125 57 29 101 14 81 137 69 41 113)(3 94 126 58 30 102 15 82 138 70 42 114)(4 95 127 59 31 103 16 83 139 71 43 115)(5 96 128 60 32 104 17 84 140 72 44 116)(6 73 129 61 33 105 18 85 141 49 45 117)(7 74 130 62 34 106 19 86 142 50 46 118)(8 75 131 63 35 107 20 87 143 51 47 119)(9 76 132 64 36 108 21 88 144 52 48 120)(10 77 133 65 37 109 22 89 121 53 25 97)(11 78 134 66 38 110 23 90 122 54 26 98)(12 79 135 67 39 111 24 91 123 55 27 99)
(1 127 13 139)(2 138 14 126)(3 125 15 137)(4 136 16 124)(5 123 17 135)(6 134 18 122)(7 121 19 133)(8 132 20 144)(9 143 21 131)(10 130 22 142)(11 141 23 129)(12 128 24 140)(25 34 37 46)(26 45 38 33)(27 32 39 44)(28 43 40 31)(29 30 41 42)(35 48 47 36)(49 98 61 110)(50 109 62 97)(51 120 63 108)(52 107 64 119)(53 118 65 106)(54 105 66 117)(55 116 67 104)(56 103 68 115)(57 114 69 102)(58 101 70 113)(59 112 71 100)(60 99 72 111)(73 90 85 78)(74 77 86 89)(75 88 87 76)(79 84 91 96)(80 95 92 83)(81 82 93 94)
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), (1,92,124,56,28,100,13,80,136,68,40,112)(2,93,125,57,29,101,14,81,137,69,41,113)(3,94,126,58,30,102,15,82,138,70,42,114)(4,95,127,59,31,103,16,83,139,71,43,115)(5,96,128,60,32,104,17,84,140,72,44,116)(6,73,129,61,33,105,18,85,141,49,45,117)(7,74,130,62,34,106,19,86,142,50,46,118)(8,75,131,63,35,107,20,87,143,51,47,119)(9,76,132,64,36,108,21,88,144,52,48,120)(10,77,133,65,37,109,22,89,121,53,25,97)(11,78,134,66,38,110,23,90,122,54,26,98)(12,79,135,67,39,111,24,91,123,55,27,99), (1,127,13,139)(2,138,14,126)(3,125,15,137)(4,136,16,124)(5,123,17,135)(6,134,18,122)(7,121,19,133)(8,132,20,144)(9,143,21,131)(10,130,22,142)(11,141,23,129)(12,128,24,140)(25,34,37,46)(26,45,38,33)(27,32,39,44)(28,43,40,31)(29,30,41,42)(35,48,47,36)(49,98,61,110)(50,109,62,97)(51,120,63,108)(52,107,64,119)(53,118,65,106)(54,105,66,117)(55,116,67,104)(56,103,68,115)(57,114,69,102)(58,101,70,113)(59,112,71,100)(60,99,72,111)(73,90,85,78)(74,77,86,89)(75,88,87,76)(79,84,91,96)(80,95,92,83)(81,82,93,94)>;
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), (1,92,124,56,28,100,13,80,136,68,40,112)(2,93,125,57,29,101,14,81,137,69,41,113)(3,94,126,58,30,102,15,82,138,70,42,114)(4,95,127,59,31,103,16,83,139,71,43,115)(5,96,128,60,32,104,17,84,140,72,44,116)(6,73,129,61,33,105,18,85,141,49,45,117)(7,74,130,62,34,106,19,86,142,50,46,118)(8,75,131,63,35,107,20,87,143,51,47,119)(9,76,132,64,36,108,21,88,144,52,48,120)(10,77,133,65,37,109,22,89,121,53,25,97)(11,78,134,66,38,110,23,90,122,54,26,98)(12,79,135,67,39,111,24,91,123,55,27,99), (1,127,13,139)(2,138,14,126)(3,125,15,137)(4,136,16,124)(5,123,17,135)(6,134,18,122)(7,121,19,133)(8,132,20,144)(9,143,21,131)(10,130,22,142)(11,141,23,129)(12,128,24,140)(25,34,37,46)(26,45,38,33)(27,32,39,44)(28,43,40,31)(29,30,41,42)(35,48,47,36)(49,98,61,110)(50,109,62,97)(51,120,63,108)(52,107,64,119)(53,118,65,106)(54,105,66,117)(55,116,67,104)(56,103,68,115)(57,114,69,102)(58,101,70,113)(59,112,71,100)(60,99,72,111)(73,90,85,78)(74,77,86,89)(75,88,87,76)(79,84,91,96)(80,95,92,83)(81,82,93,94) );
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)], [(1,92,124,56,28,100,13,80,136,68,40,112),(2,93,125,57,29,101,14,81,137,69,41,113),(3,94,126,58,30,102,15,82,138,70,42,114),(4,95,127,59,31,103,16,83,139,71,43,115),(5,96,128,60,32,104,17,84,140,72,44,116),(6,73,129,61,33,105,18,85,141,49,45,117),(7,74,130,62,34,106,19,86,142,50,46,118),(8,75,131,63,35,107,20,87,143,51,47,119),(9,76,132,64,36,108,21,88,144,52,48,120),(10,77,133,65,37,109,22,89,121,53,25,97),(11,78,134,66,38,110,23,90,122,54,26,98),(12,79,135,67,39,111,24,91,123,55,27,99)], [(1,127,13,139),(2,138,14,126),(3,125,15,137),(4,136,16,124),(5,123,17,135),(6,134,18,122),(7,121,19,133),(8,132,20,144),(9,143,21,131),(10,130,22,142),(11,141,23,129),(12,128,24,140),(25,34,37,46),(26,45,38,33),(27,32,39,44),(28,43,40,31),(29,30,41,42),(35,48,47,36),(49,98,61,110),(50,109,62,97),(51,120,63,108),(52,107,64,119),(53,118,65,106),(54,105,66,117),(55,116,67,104),(56,103,68,115),(57,114,69,102),(58,101,70,113),(59,112,71,100),(60,99,72,111),(73,90,85,78),(74,77,86,89),(75,88,87,76),(79,84,91,96),(80,95,92,83),(81,82,93,94)]])
78 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3A | 3B | 3C | 3D | 4A | 4B | 4C | 4D | 4E | 6A | ··· | 6L | 8A | 8B | 8C | 8D | 12A | ··· | 12P | 24A | ··· | 24AF |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 2 | 36 | 36 | 2 | 2 | 2 | 2 | 1 | 1 | 2 | 36 | 36 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
78 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | D12 | D12 | C4○D8 | C4○D24 |
| kernel | C24.78D6 | C24⋊2S3 | C32⋊5D8 | C32⋊5Q16 | C6×C24 | C12.59D6 | C2×C24 | C3×C12 | C62 | C24 | C2×C12 | C12 | C2×C6 | C32 | C3 |
| # reps | 1 | 2 | 1 | 1 | 1 | 2 | 4 | 1 | 1 | 8 | 4 | 8 | 8 | 4 | 32 |
Matrix representation of C24.78D6 ►in GL4(𝔽73) generated by
| 0 | 1 | 0 | 0 |
| 72 | 72 | 0 | 0 |
| 0 | 0 | 25 | 36 |
| 0 | 0 | 37 | 62 |
| 0 | 72 | 0 | 0 |
| 1 | 1 | 0 | 0 |
| 0 | 0 | 27 | 0 |
| 0 | 0 | 0 | 27 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 25 | 36 |
| 0 | 0 | 11 | 48 |
G:=sub<GL(4,GF(73))| [0,72,0,0,1,72,0,0,0,0,25,37,0,0,36,62],[0,1,0,0,72,1,0,0,0,0,27,0,0,0,0,27],[0,1,0,0,1,0,0,0,0,0,25,11,0,0,36,48] >;
C24.78D6 in GAP, Magma, Sage, TeX
C_{24}._{78}D_6 % in TeX
G:=Group("C24.78D6"); // GroupNames label
G:=SmallGroup(288,761);
// by ID
G=gap.SmallGroup(288,761);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,120,254,58,675,80,2693,9414]);
// Polycyclic
G:=Group<a,b,c|a^24=1,b^6=c^2=a^12,a*b=b*a,c*a*c^-1=a^11,c*b*c^-1=b^5>;
// generators/relations