metabelian, supersoluble, monomial
Aliases: C24.5D6, C12.33D12, C62.71D4, C24⋊2S3⋊4C2, C6.61(C2×D12), (C3×C12).96D4, (C2×C6).18D12, C32⋊5Q16⋊3C2, (C2×C12).148D6, C3⋊3(C8.D6), (C3×M4(2))⋊2S3, (C3×C24).5C22, M4(2)⋊2(C3⋊S3), C4.15(C12⋊S3), (C6×C12).139C22, C12.194(C22×S3), (C3×C12).156C23, C12.59D6.7C2, C32⋊17(C8.C22), (C32×M4(2))⋊4C2, C12⋊S3.24C22, C22.6(C12⋊S3), C32⋊4Q8.25C22, C8.1(C2×C3⋊S3), (C3×C6).201(C2×D4), C4.31(C22×C3⋊S3), C2.16(C2×C12⋊S3), (C2×C32⋊4Q8)⋊14C2, (C2×C4).16(C2×C3⋊S3), SmallGroup(288,766)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24.5D6
G = < a,b,c | a24=1, b6=c2=a12, bab-1=a13, cac-1=a-1, cbc-1=b5 >
Subgroups: 748 in 180 conjugacy classes, 65 normal (19 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, M4(2), SD16, Q16, C2×Q8, C4○D4, C3⋊S3, C3×C6, C3×C6, C24, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C8.C22, C3⋊Dic3, C3×C12, C2×C3⋊S3, C62, C24⋊C2, Dic12, C3×M4(2), C2×Dic6, C4○D12, C3×C24, C32⋊4Q8, C32⋊4Q8, C32⋊4Q8, C4×C3⋊S3, C12⋊S3, C2×C3⋊Dic3, C32⋊7D4, C6×C12, C8.D6, C24⋊2S3, C32⋊5Q16, C32×M4(2), C2×C32⋊4Q8, C12.59D6, C24.5D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C3⋊S3, D12, C22×S3, C8.C22, C2×C3⋊S3, C2×D12, C12⋊S3, C22×C3⋊S3, C8.D6, C2×C12⋊S3, C24.5D6
►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 26 136 77 70 102 13 38 124 89 58 114)(2 39 137 90 71 115 14 27 125 78 59 103)(3 28 138 79 72 104 15 40 126 91 60 116)(4 41 139 92 49 117 16 29 127 80 61 105)(5 30 140 81 50 106 17 42 128 93 62 118)(6 43 141 94 51 119 18 31 129 82 63 107)(7 32 142 83 52 108 19 44 130 95 64 120)(8 45 143 96 53 97 20 33 131 84 65 109)(9 34 144 85 54 110 21 46 132 73 66 98)(10 47 121 74 55 99 22 35 133 86 67 111)(11 36 122 87 56 112 23 48 134 75 68 100)(12 25 123 76 57 101 24 37 135 88 69 113)
(1 102 13 114)(2 101 14 113)(3 100 15 112)(4 99 16 111)(5 98 17 110)(6 97 18 109)(7 120 19 108)(8 119 20 107)(9 118 21 106)(10 117 22 105)(11 116 23 104)(12 115 24 103)(25 59 37 71)(26 58 38 70)(27 57 39 69)(28 56 40 68)(29 55 41 67)(30 54 42 66)(31 53 43 65)(32 52 44 64)(33 51 45 63)(34 50 46 62)(35 49 47 61)(36 72 48 60)(73 128 85 140)(74 127 86 139)(75 126 87 138)(76 125 88 137)(77 124 89 136)(78 123 90 135)(79 122 91 134)(80 121 92 133)(81 144 93 132)(82 143 94 131)(83 142 95 130)(84 141 96 129)
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), (1,26,136,77,70,102,13,38,124,89,58,114)(2,39,137,90,71,115,14,27,125,78,59,103)(3,28,138,79,72,104,15,40,126,91,60,116)(4,41,139,92,49,117,16,29,127,80,61,105)(5,30,140,81,50,106,17,42,128,93,62,118)(6,43,141,94,51,119,18,31,129,82,63,107)(7,32,142,83,52,108,19,44,130,95,64,120)(8,45,143,96,53,97,20,33,131,84,65,109)(9,34,144,85,54,110,21,46,132,73,66,98)(10,47,121,74,55,99,22,35,133,86,67,111)(11,36,122,87,56,112,23,48,134,75,68,100)(12,25,123,76,57,101,24,37,135,88,69,113), (1,102,13,114)(2,101,14,113)(3,100,15,112)(4,99,16,111)(5,98,17,110)(6,97,18,109)(7,120,19,108)(8,119,20,107)(9,118,21,106)(10,117,22,105)(11,116,23,104)(12,115,24,103)(25,59,37,71)(26,58,38,70)(27,57,39,69)(28,56,40,68)(29,55,41,67)(30,54,42,66)(31,53,43,65)(32,52,44,64)(33,51,45,63)(34,50,46,62)(35,49,47,61)(36,72,48,60)(73,128,85,140)(74,127,86,139)(75,126,87,138)(76,125,88,137)(77,124,89,136)(78,123,90,135)(79,122,91,134)(80,121,92,133)(81,144,93,132)(82,143,94,131)(83,142,95,130)(84,141,96,129)>;
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), (1,26,136,77,70,102,13,38,124,89,58,114)(2,39,137,90,71,115,14,27,125,78,59,103)(3,28,138,79,72,104,15,40,126,91,60,116)(4,41,139,92,49,117,16,29,127,80,61,105)(5,30,140,81,50,106,17,42,128,93,62,118)(6,43,141,94,51,119,18,31,129,82,63,107)(7,32,142,83,52,108,19,44,130,95,64,120)(8,45,143,96,53,97,20,33,131,84,65,109)(9,34,144,85,54,110,21,46,132,73,66,98)(10,47,121,74,55,99,22,35,133,86,67,111)(11,36,122,87,56,112,23,48,134,75,68,100)(12,25,123,76,57,101,24,37,135,88,69,113), (1,102,13,114)(2,101,14,113)(3,100,15,112)(4,99,16,111)(5,98,17,110)(6,97,18,109)(7,120,19,108)(8,119,20,107)(9,118,21,106)(10,117,22,105)(11,116,23,104)(12,115,24,103)(25,59,37,71)(26,58,38,70)(27,57,39,69)(28,56,40,68)(29,55,41,67)(30,54,42,66)(31,53,43,65)(32,52,44,64)(33,51,45,63)(34,50,46,62)(35,49,47,61)(36,72,48,60)(73,128,85,140)(74,127,86,139)(75,126,87,138)(76,125,88,137)(77,124,89,136)(78,123,90,135)(79,122,91,134)(80,121,92,133)(81,144,93,132)(82,143,94,131)(83,142,95,130)(84,141,96,129) );
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)], [(1,26,136,77,70,102,13,38,124,89,58,114),(2,39,137,90,71,115,14,27,125,78,59,103),(3,28,138,79,72,104,15,40,126,91,60,116),(4,41,139,92,49,117,16,29,127,80,61,105),(5,30,140,81,50,106,17,42,128,93,62,118),(6,43,141,94,51,119,18,31,129,82,63,107),(7,32,142,83,52,108,19,44,130,95,64,120),(8,45,143,96,53,97,20,33,131,84,65,109),(9,34,144,85,54,110,21,46,132,73,66,98),(10,47,121,74,55,99,22,35,133,86,67,111),(11,36,122,87,56,112,23,48,134,75,68,100),(12,25,123,76,57,101,24,37,135,88,69,113)], [(1,102,13,114),(2,101,14,113),(3,100,15,112),(4,99,16,111),(5,98,17,110),(6,97,18,109),(7,120,19,108),(8,119,20,107),(9,118,21,106),(10,117,22,105),(11,116,23,104),(12,115,24,103),(25,59,37,71),(26,58,38,70),(27,57,39,69),(28,56,40,68),(29,55,41,67),(30,54,42,66),(31,53,43,65),(32,52,44,64),(33,51,45,63),(34,50,46,62),(35,49,47,61),(36,72,48,60),(73,128,85,140),(74,127,86,139),(75,126,87,138),(76,125,88,137),(77,124,89,136),(78,123,90,135),(79,122,91,134),(80,121,92,133),(81,144,93,132),(82,143,94,131),(83,142,95,130),(84,141,96,129)]])
51 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 8A | 8B | 12A | ··· | 12H | 12I | 12J | 12K | 12L | 24A | ··· | 24P |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 12 | ··· | 12 | 12 | 12 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 2 | 36 | 2 | 2 | 2 | 2 | 2 | 2 | 36 | 36 | 36 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
51 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - |
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | D12 | D12 | C8.C22 | C8.D6 |
| kernel | C24.5D6 | C24⋊2S3 | C32⋊5Q16 | C32×M4(2) | C2×C32⋊4Q8 | C12.59D6 | C3×M4(2) | C3×C12 | C62 | C24 | C2×C12 | C12 | C2×C6 | C32 | C3 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 4 | 1 | 1 | 8 | 4 | 8 | 8 | 1 | 8 |
Matrix representation of C24.5D6 ►in GL6(𝔽73)
| 72 | 72 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 71 | 38 | 0 | 26 |
| 0 | 0 | 35 | 33 | 47 | 47 |
| 0 | 0 | 56 | 37 | 2 | 35 |
| 0 | 0 | 36 | 19 | 38 | 40 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 45 | 43 | 40 |
| 0 | 0 | 28 | 35 | 33 | 3 |
| 0 | 0 | 57 | 21 | 66 | 28 |
| 0 | 0 | 52 | 36 | 45 | 38 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 45 | 43 | 40 |
| 0 | 0 | 38 | 66 | 70 | 30 |
| 0 | 0 | 34 | 52 | 2 | 3 |
| 0 | 0 | 18 | 39 | 1 | 71 |
G:=sub<GL(6,GF(73))| [72,1,0,0,0,0,72,0,0,0,0,0,0,0,71,35,56,36,0,0,38,33,37,19,0,0,0,47,2,38,0,0,26,47,35,40],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,7,28,57,52,0,0,45,35,21,36,0,0,43,33,66,45,0,0,40,3,28,38],[72,1,0,0,0,0,0,1,0,0,0,0,0,0,7,38,34,18,0,0,45,66,52,39,0,0,43,70,2,1,0,0,40,30,3,71] >;
C24.5D6 in GAP, Magma, Sage, TeX
C_{24}._5D_6 % in TeX
G:=Group("C24.5D6"); // GroupNames label
G:=SmallGroup(288,766);
// by ID
G=gap.SmallGroup(288,766);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,120,254,219,58,675,80,2693,9414]);
// Polycyclic
G:=Group<a,b,c|a^24=1,b^6=c^2=a^12,b*a*b^-1=a^13,c*a*c^-1=a^-1,c*b*c^-1=b^5>;
// generators/relations