metabelian, supersoluble, monomial
Aliases: C24.35D6, (C3×Q16)⋊4S3, C24⋊2S3⋊6C2, C24⋊S3⋊6C2, Q16⋊2(C3⋊S3), C6.126(S3×D4), (C3×Q8).39D6, C3⋊5(Q16⋊S3), C3⋊Dic3.68D4, (C32×Q16)⋊7C2, C32⋊7Q16⋊8C2, C32⋊11SD16⋊7C2, (C3×C12).99C23, (C3×C24).34C22, C12.95(C22×S3), C12.26D6.3C2, C32⋊21(C8.C22), C12⋊S3.19C22, C32⋊4C8.16C22, (Q8×C32).19C22, C32⋊4Q8.20C22, C8.3(C2×C3⋊S3), (Q8×C3⋊S3)⋊5C2, C2.23(D4×C3⋊S3), Q8.9(C2×C3⋊S3), (C2×C3⋊S3).68D4, C4.9(C22×C3⋊S3), (C3×C6).247(C2×D4), (C4×C3⋊S3).26C22, SmallGroup(288,775)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24.35D6
G = < a,b,c | a24=1, b6=c2=a12, bab-1=a7, cac-1=a11, cbc-1=b5 >
Subgroups: 748 in 180 conjugacy classes, 55 normal (27 characteristic)
C1, C2, C2, C3, C4, C4, C22, S3, C6, C8, C8, C2×C4, D4, Q8, Q8, C32, Dic3, C12, C12, D6, M4(2), SD16, Q16, Q16, C2×Q8, C4○D4, C3⋊S3, C3×C6, C3⋊C8, C24, Dic6, C4×S3, D12, C3×Q8, C8.C22, C3⋊Dic3, C3⋊Dic3, C3×C12, C3×C12, C2×C3⋊S3, C2×C3⋊S3, C8⋊S3, C24⋊C2, Q8⋊2S3, C3⋊Q16, C3×Q16, S3×Q8, Q8⋊3S3, C32⋊4C8, C3×C24, C32⋊4Q8, C32⋊4Q8, C4×C3⋊S3, C4×C3⋊S3, C12⋊S3, C12⋊S3, Q8×C32, Q16⋊S3, C24⋊S3, C24⋊2S3, C32⋊11SD16, C32⋊7Q16, C32×Q16, Q8×C3⋊S3, C12.26D6, C24.35D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C3⋊S3, C22×S3, C8.C22, C2×C3⋊S3, S3×D4, C22×C3⋊S3, Q16⋊S3, D4×C3⋊S3, C24.35D6
►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 76 129 36 54 116 13 88 141 48 66 104)(2 83 130 43 55 99 14 95 142 31 67 111)(3 90 131 26 56 106 15 78 143 38 68 118)(4 73 132 33 57 113 16 85 144 45 69 101)(5 80 133 40 58 120 17 92 121 28 70 108)(6 87 134 47 59 103 18 75 122 35 71 115)(7 94 135 30 60 110 19 82 123 42 72 98)(8 77 136 37 61 117 20 89 124 25 49 105)(9 84 137 44 62 100 21 96 125 32 50 112)(10 91 138 27 63 107 22 79 126 39 51 119)(11 74 139 34 64 114 23 86 127 46 52 102)(12 81 140 41 65 97 24 93 128 29 53 109)
(1 144 13 132)(2 131 14 143)(3 142 15 130)(4 129 16 141)(5 140 17 128)(6 127 18 139)(7 138 19 126)(8 125 20 137)(9 136 21 124)(10 123 22 135)(11 134 23 122)(12 121 24 133)(25 100 37 112)(26 111 38 99)(27 98 39 110)(28 109 40 97)(29 120 41 108)(30 107 42 119)(31 118 43 106)(32 105 44 117)(33 116 45 104)(34 103 46 115)(35 114 47 102)(36 101 48 113)(49 50 61 62)(51 72 63 60)(52 59 64 71)(53 70 65 58)(54 57 66 69)(55 68 67 56)(73 88 85 76)(74 75 86 87)(77 84 89 96)(78 95 90 83)(79 82 91 94)(80 93 92 81)
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), (1,76,129,36,54,116,13,88,141,48,66,104)(2,83,130,43,55,99,14,95,142,31,67,111)(3,90,131,26,56,106,15,78,143,38,68,118)(4,73,132,33,57,113,16,85,144,45,69,101)(5,80,133,40,58,120,17,92,121,28,70,108)(6,87,134,47,59,103,18,75,122,35,71,115)(7,94,135,30,60,110,19,82,123,42,72,98)(8,77,136,37,61,117,20,89,124,25,49,105)(9,84,137,44,62,100,21,96,125,32,50,112)(10,91,138,27,63,107,22,79,126,39,51,119)(11,74,139,34,64,114,23,86,127,46,52,102)(12,81,140,41,65,97,24,93,128,29,53,109), (1,144,13,132)(2,131,14,143)(3,142,15,130)(4,129,16,141)(5,140,17,128)(6,127,18,139)(7,138,19,126)(8,125,20,137)(9,136,21,124)(10,123,22,135)(11,134,23,122)(12,121,24,133)(25,100,37,112)(26,111,38,99)(27,98,39,110)(28,109,40,97)(29,120,41,108)(30,107,42,119)(31,118,43,106)(32,105,44,117)(33,116,45,104)(34,103,46,115)(35,114,47,102)(36,101,48,113)(49,50,61,62)(51,72,63,60)(52,59,64,71)(53,70,65,58)(54,57,66,69)(55,68,67,56)(73,88,85,76)(74,75,86,87)(77,84,89,96)(78,95,90,83)(79,82,91,94)(80,93,92,81)>;
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), (1,76,129,36,54,116,13,88,141,48,66,104)(2,83,130,43,55,99,14,95,142,31,67,111)(3,90,131,26,56,106,15,78,143,38,68,118)(4,73,132,33,57,113,16,85,144,45,69,101)(5,80,133,40,58,120,17,92,121,28,70,108)(6,87,134,47,59,103,18,75,122,35,71,115)(7,94,135,30,60,110,19,82,123,42,72,98)(8,77,136,37,61,117,20,89,124,25,49,105)(9,84,137,44,62,100,21,96,125,32,50,112)(10,91,138,27,63,107,22,79,126,39,51,119)(11,74,139,34,64,114,23,86,127,46,52,102)(12,81,140,41,65,97,24,93,128,29,53,109), (1,144,13,132)(2,131,14,143)(3,142,15,130)(4,129,16,141)(5,140,17,128)(6,127,18,139)(7,138,19,126)(8,125,20,137)(9,136,21,124)(10,123,22,135)(11,134,23,122)(12,121,24,133)(25,100,37,112)(26,111,38,99)(27,98,39,110)(28,109,40,97)(29,120,41,108)(30,107,42,119)(31,118,43,106)(32,105,44,117)(33,116,45,104)(34,103,46,115)(35,114,47,102)(36,101,48,113)(49,50,61,62)(51,72,63,60)(52,59,64,71)(53,70,65,58)(54,57,66,69)(55,68,67,56)(73,88,85,76)(74,75,86,87)(77,84,89,96)(78,95,90,83)(79,82,91,94)(80,93,92,81) );
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)], [(1,76,129,36,54,116,13,88,141,48,66,104),(2,83,130,43,55,99,14,95,142,31,67,111),(3,90,131,26,56,106,15,78,143,38,68,118),(4,73,132,33,57,113,16,85,144,45,69,101),(5,80,133,40,58,120,17,92,121,28,70,108),(6,87,134,47,59,103,18,75,122,35,71,115),(7,94,135,30,60,110,19,82,123,42,72,98),(8,77,136,37,61,117,20,89,124,25,49,105),(9,84,137,44,62,100,21,96,125,32,50,112),(10,91,138,27,63,107,22,79,126,39,51,119),(11,74,139,34,64,114,23,86,127,46,52,102),(12,81,140,41,65,97,24,93,128,29,53,109)], [(1,144,13,132),(2,131,14,143),(3,142,15,130),(4,129,16,141),(5,140,17,128),(6,127,18,139),(7,138,19,126),(8,125,20,137),(9,136,21,124),(10,123,22,135),(11,134,23,122),(12,121,24,133),(25,100,37,112),(26,111,38,99),(27,98,39,110),(28,109,40,97),(29,120,41,108),(30,107,42,119),(31,118,43,106),(32,105,44,117),(33,116,45,104),(34,103,46,115),(35,114,47,102),(36,101,48,113),(49,50,61,62),(51,72,63,60),(52,59,64,71),(53,70,65,58),(54,57,66,69),(55,68,67,56),(73,88,85,76),(74,75,86,87),(77,84,89,96),(78,95,90,83),(79,82,91,94),(80,93,92,81)]])
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 6C | 6D | 8A | 8B | 12A | 12B | 12C | 12D | 12E | ··· | 12L | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 18 | 36 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 18 | 36 | 2 | 2 | 2 | 2 | 4 | 36 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | C8.C22 | S3×D4 | Q16⋊S3 |
| kernel | C24.35D6 | C24⋊S3 | C24⋊2S3 | C32⋊11SD16 | C32⋊7Q16 | C32×Q16 | Q8×C3⋊S3 | C12.26D6 | C3×Q16 | C3⋊Dic3 | C2×C3⋊S3 | C24 | C3×Q8 | C32 | C6 | C3 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 4 | 8 | 1 | 4 | 8 |
Matrix representation of C24.35D6 ►in GL8(𝔽73)
| 1 | 37 | 30 | 15 | 0 | 0 | 0 | 0 |
| 36 | 37 | 58 | 15 | 0 | 0 | 0 | 0 |
| 43 | 58 | 72 | 36 | 0 | 0 | 0 | 0 |
| 15 | 58 | 37 | 36 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 22 | 0 | 20 |
| 0 | 0 | 0 | 0 | 51 | 62 | 53 | 53 |
| 0 | 0 | 0 | 0 | 11 | 11 | 0 | 0 |
| 0 | 0 | 0 | 0 | 62 | 0 | 0 | 0 |
| 15 | 58 | 37 | 36 | 0 | 0 | 0 | 0 |
| 15 | 30 | 37 | 1 | 0 | 0 | 0 | 0 |
| 36 | 37 | 58 | 15 | 0 | 0 | 0 | 0 |
| 36 | 72 | 58 | 43 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 37 | 37 | 57 | 36 |
| 0 | 0 | 0 | 0 | 36 | 0 | 37 | 21 |
| 0 | 0 | 0 | 0 | 49 | 35 | 36 | 36 |
| 0 | 0 | 0 | 0 | 38 | 14 | 37 | 0 |
| 36 | 37 | 58 | 15 | 0 | 0 | 0 | 0 |
| 1 | 37 | 30 | 15 | 0 | 0 | 0 | 0 |
| 15 | 58 | 37 | 36 | 0 | 0 | 0 | 0 |
| 43 | 58 | 72 | 36 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 62 | 51 | 0 | 53 |
| 0 | 0 | 0 | 0 | 62 | 11 | 53 | 0 |
| 0 | 0 | 0 | 0 | 0 | 62 | 62 | 11 |
| 0 | 0 | 0 | 0 | 62 | 0 | 22 | 11 |
G:=sub<GL(8,GF(73))| [1,36,43,15,0,0,0,0,37,37,58,58,0,0,0,0,30,58,72,37,0,0,0,0,15,15,36,36,0,0,0,0,0,0,0,0,11,51,11,62,0,0,0,0,22,62,11,0,0,0,0,0,0,53,0,0,0,0,0,0,20,53,0,0],[15,15,36,36,0,0,0,0,58,30,37,72,0,0,0,0,37,37,58,58,0,0,0,0,36,1,15,43,0,0,0,0,0,0,0,0,37,36,49,38,0,0,0,0,37,0,35,14,0,0,0,0,57,37,36,37,0,0,0,0,36,21,36,0],[36,1,15,43,0,0,0,0,37,37,58,58,0,0,0,0,58,30,37,72,0,0,0,0,15,15,36,36,0,0,0,0,0,0,0,0,62,62,0,62,0,0,0,0,51,11,62,0,0,0,0,0,0,53,62,22,0,0,0,0,53,0,11,11] >;
C24.35D6 in GAP, Magma, Sage, TeX
C_{24}._{35}D_6 % in TeX
G:=Group("C24.35D6"); // GroupNames label
G:=SmallGroup(288,775);
// by ID
G=gap.SmallGroup(288,775);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,120,422,135,100,346,185,80,2693,9414]);
// Polycyclic
G:=Group<a,b,c|a^24=1,b^6=c^2=a^12,b*a*b^-1=a^7,c*a*c^-1=a^11,c*b*c^-1=b^5>;
// generators/relations