metabelian, supersoluble, monomial
Aliases: C24.40D6, (C3×SD16)⋊4S3, (C3×D4).19D6, C6.124(S3×D4), C24⋊2S3⋊10C2, C32⋊7D8⋊8C2, (C3×Q8).37D6, SD16⋊3(C3⋊S3), C32⋊21(C4○D8), C3⋊Dic3.70D4, C32⋊7Q16⋊6C2, C3⋊5(Q8.7D6), C12.26D6⋊4C2, C12.D6⋊5C2, (C3×C12).97C23, C12.93(C22×S3), (C3×C24).39C22, (C32×SD16)⋊8C2, C12⋊S3.18C22, C32⋊4C8.27C22, (D4×C32).20C22, (Q8×C32).17C22, C32⋊4Q8.18C22, (C8×C3⋊S3)⋊9C2, C8.11(C2×C3⋊S3), D4.5(C2×C3⋊S3), C2.21(D4×C3⋊S3), Q8.7(C2×C3⋊S3), (C2×C3⋊S3).47D4, C4.7(C22×C3⋊S3), (C3×C6).245(C2×D4), (C4×C3⋊S3).73C22, SmallGroup(288,773)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24.40D6
G = < a,b,c | a24=b6=1, c2=a12, bab-1=a19, cac-1=a11, cbc-1=a12b-1 >
Subgroups: 772 in 186 conjugacy classes, 55 normal (27 characteristic)
C1, C2, C2, C3, C4, C4, C22, S3, C6, C6, C8, C8, C2×C4, D4, D4, Q8, Q8, C32, Dic3, C12, C12, D6, C2×C6, C2×C8, D8, SD16, SD16, Q16, C4○D4, C3⋊S3, C3×C6, C3×C6, C3⋊C8, C24, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C3×D4, C3×Q8, C4○D8, C3⋊Dic3, C3⋊Dic3, C3×C12, C3×C12, C2×C3⋊S3, C2×C3⋊S3, C62, S3×C8, C24⋊C2, D4⋊S3, C3⋊Q16, C3×SD16, D4⋊2S3, Q8⋊3S3, C32⋊4C8, C3×C24, C32⋊4Q8, C4×C3⋊S3, C4×C3⋊S3, C12⋊S3, C12⋊S3, C2×C3⋊Dic3, C32⋊7D4, D4×C32, Q8×C32, Q8.7D6, C8×C3⋊S3, C24⋊2S3, C32⋊7D8, C32⋊7Q16, C32×SD16, C12.D6, C12.26D6, C24.40D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C3⋊S3, C22×S3, C4○D8, C2×C3⋊S3, S3×D4, C22×C3⋊S3, Q8.7D6, D4×C3⋊S3, C24.40D6
►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)
(1 137 93 46 64 110)(2 132 94 41 65 105)(3 127 95 36 66 100)(4 122 96 31 67 119)(5 141 73 26 68 114)(6 136 74 45 69 109)(7 131 75 40 70 104)(8 126 76 35 71 99)(9 121 77 30 72 118)(10 140 78 25 49 113)(11 135 79 44 50 108)(12 130 80 39 51 103)(13 125 81 34 52 98)(14 144 82 29 53 117)(15 139 83 48 54 112)(16 134 84 43 55 107)(17 129 85 38 56 102)(18 124 86 33 57 97)(19 143 87 28 58 116)(20 138 88 47 59 111)(21 133 89 42 60 106)(22 128 90 37 61 101)(23 123 91 32 62 120)(24 142 92 27 63 115)
(1 84 13 96)(2 95 14 83)(3 82 15 94)(4 93 16 81)(5 80 17 92)(6 91 18 79)(7 78 19 90)(8 89 20 77)(9 76 21 88)(10 87 22 75)(11 74 23 86)(12 85 24 73)(25 104 37 116)(26 115 38 103)(27 102 39 114)(28 113 40 101)(29 100 41 112)(30 111 42 99)(31 98 43 110)(32 109 44 97)(33 120 45 108)(34 107 46 119)(35 118 47 106)(36 105 48 117)(49 58 61 70)(50 69 62 57)(51 56 63 68)(52 67 64 55)(53 54 65 66)(59 72 71 60)(121 138 133 126)(122 125 134 137)(123 136 135 124)(127 132 139 144)(128 143 140 131)(129 130 141 142)
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), (1,137,93,46,64,110)(2,132,94,41,65,105)(3,127,95,36,66,100)(4,122,96,31,67,119)(5,141,73,26,68,114)(6,136,74,45,69,109)(7,131,75,40,70,104)(8,126,76,35,71,99)(9,121,77,30,72,118)(10,140,78,25,49,113)(11,135,79,44,50,108)(12,130,80,39,51,103)(13,125,81,34,52,98)(14,144,82,29,53,117)(15,139,83,48,54,112)(16,134,84,43,55,107)(17,129,85,38,56,102)(18,124,86,33,57,97)(19,143,87,28,58,116)(20,138,88,47,59,111)(21,133,89,42,60,106)(22,128,90,37,61,101)(23,123,91,32,62,120)(24,142,92,27,63,115), (1,84,13,96)(2,95,14,83)(3,82,15,94)(4,93,16,81)(5,80,17,92)(6,91,18,79)(7,78,19,90)(8,89,20,77)(9,76,21,88)(10,87,22,75)(11,74,23,86)(12,85,24,73)(25,104,37,116)(26,115,38,103)(27,102,39,114)(28,113,40,101)(29,100,41,112)(30,111,42,99)(31,98,43,110)(32,109,44,97)(33,120,45,108)(34,107,46,119)(35,118,47,106)(36,105,48,117)(49,58,61,70)(50,69,62,57)(51,56,63,68)(52,67,64,55)(53,54,65,66)(59,72,71,60)(121,138,133,126)(122,125,134,137)(123,136,135,124)(127,132,139,144)(128,143,140,131)(129,130,141,142)>;
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), (1,137,93,46,64,110)(2,132,94,41,65,105)(3,127,95,36,66,100)(4,122,96,31,67,119)(5,141,73,26,68,114)(6,136,74,45,69,109)(7,131,75,40,70,104)(8,126,76,35,71,99)(9,121,77,30,72,118)(10,140,78,25,49,113)(11,135,79,44,50,108)(12,130,80,39,51,103)(13,125,81,34,52,98)(14,144,82,29,53,117)(15,139,83,48,54,112)(16,134,84,43,55,107)(17,129,85,38,56,102)(18,124,86,33,57,97)(19,143,87,28,58,116)(20,138,88,47,59,111)(21,133,89,42,60,106)(22,128,90,37,61,101)(23,123,91,32,62,120)(24,142,92,27,63,115), (1,84,13,96)(2,95,14,83)(3,82,15,94)(4,93,16,81)(5,80,17,92)(6,91,18,79)(7,78,19,90)(8,89,20,77)(9,76,21,88)(10,87,22,75)(11,74,23,86)(12,85,24,73)(25,104,37,116)(26,115,38,103)(27,102,39,114)(28,113,40,101)(29,100,41,112)(30,111,42,99)(31,98,43,110)(32,109,44,97)(33,120,45,108)(34,107,46,119)(35,118,47,106)(36,105,48,117)(49,58,61,70)(50,69,62,57)(51,56,63,68)(52,67,64,55)(53,54,65,66)(59,72,71,60)(121,138,133,126)(122,125,134,137)(123,136,135,124)(127,132,139,144)(128,143,140,131)(129,130,141,142) );
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)], [(1,137,93,46,64,110),(2,132,94,41,65,105),(3,127,95,36,66,100),(4,122,96,31,67,119),(5,141,73,26,68,114),(6,136,74,45,69,109),(7,131,75,40,70,104),(8,126,76,35,71,99),(9,121,77,30,72,118),(10,140,78,25,49,113),(11,135,79,44,50,108),(12,130,80,39,51,103),(13,125,81,34,52,98),(14,144,82,29,53,117),(15,139,83,48,54,112),(16,134,84,43,55,107),(17,129,85,38,56,102),(18,124,86,33,57,97),(19,143,87,28,58,116),(20,138,88,47,59,111),(21,133,89,42,60,106),(22,128,90,37,61,101),(23,123,91,32,62,120),(24,142,92,27,63,115)], [(1,84,13,96),(2,95,14,83),(3,82,15,94),(4,93,16,81),(5,80,17,92),(6,91,18,79),(7,78,19,90),(8,89,20,77),(9,76,21,88),(10,87,22,75),(11,74,23,86),(12,85,24,73),(25,104,37,116),(26,115,38,103),(27,102,39,114),(28,113,40,101),(29,100,41,112),(30,111,42,99),(31,98,43,110),(32,109,44,97),(33,120,45,108),(34,107,46,119),(35,118,47,106),(36,105,48,117),(49,58,61,70),(50,69,62,57),(51,56,63,68),(52,67,64,55),(53,54,65,66),(59,72,71,60),(121,138,133,126),(122,125,134,137),(123,136,135,124),(127,132,139,144),(128,143,140,131),(129,130,141,142)]])
42 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 | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 24A | ··· | 24H |
| 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 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 4 | 18 | 36 | 2 | 2 | 2 | 2 | 2 | 4 | 9 | 9 | 36 | 2 | 2 | 2 | 2 | 8 | 8 | 8 | 8 | 2 | 2 | 18 | 18 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | D6 | C4○D8 | S3×D4 | Q8.7D6 |
| kernel | C24.40D6 | C8×C3⋊S3 | C24⋊2S3 | C32⋊7D8 | C32⋊7Q16 | C32×SD16 | C12.D6 | C12.26D6 | C3×SD16 | C3⋊Dic3 | C2×C3⋊S3 | C24 | C3×D4 | C3×Q8 | C32 | C6 | C3 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 4 | 4 | 4 | 4 | 4 | 8 |
Matrix representation of C24.40D6 ►in GL6(𝔽73)
| 0 | 1 | 0 | 0 | 0 | 0 |
| 72 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 72 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 69 |
| 0 | 0 | 0 | 0 | 18 | 61 |
| 72 | 72 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 1 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 32 | 38 |
| 0 | 0 | 0 | 0 | 48 | 41 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 72 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 72 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 0 | 0 | 0 | 0 | 18 | 0 |
G:=sub<GL(6,GF(73))| [0,72,0,0,0,0,1,72,0,0,0,0,0,0,0,72,0,0,0,0,1,1,0,0,0,0,0,0,0,18,0,0,0,0,69,61],[72,1,0,0,0,0,72,0,0,0,0,0,0,0,0,1,0,0,0,0,72,72,0,0,0,0,0,0,32,48,0,0,0,0,38,41],[1,72,0,0,0,0,0,72,0,0,0,0,0,0,72,72,0,0,0,0,0,1,0,0,0,0,0,0,0,18,0,0,0,0,4,0] >;
C24.40D6 in GAP, Magma, Sage, TeX
C_{24}._{40}D_6 % in TeX
G:=Group("C24.40D6"); // GroupNames label
G:=SmallGroup(288,773);
// by ID
G=gap.SmallGroup(288,773);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,422,135,100,346,185,80,2693,9414]);
// Polycyclic
G:=Group<a,b,c|a^24=b^6=1,c^2=a^12,b*a*b^-1=a^19,c*a*c^-1=a^11,c*b*c^-1=a^12*b^-1>;
// generators/relations