metabelian, supersoluble, monomial
Aliases: C24.32D6, C24⋊S3⋊4C2, (C3×SD16)⋊2S3, (C3×D4).18D6, C6.123(S3×D4), (C3×Q8).36D6, SD16⋊2(C3⋊S3), C3⋊5(D4.D6), C3⋊Dic3.67D4, C32⋊7Q16⋊5C2, C32⋊5Q16⋊10C2, C32⋊9SD16⋊8C2, (C3×C12).96C23, C12.92(C22×S3), (C3×C24).33C22, (C32×SD16)⋊4C2, C12.D6.3C2, C32⋊19(C8.C22), C32⋊4C8.15C22, (D4×C32).19C22, (Q8×C32).16C22, C32⋊4Q8.17C22, C8.2(C2×C3⋊S3), (Q8×C3⋊S3)⋊4C2, D4.4(C2×C3⋊S3), C2.20(D4×C3⋊S3), Q8.6(C2×C3⋊S3), (C2×C3⋊S3).67D4, C4.6(C22×C3⋊S3), (C3×C6).244(C2×D4), (C4×C3⋊S3).25C22, SmallGroup(288,772)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24.32D6
G = < a,b,c | a24=1, b6=c2=a12, bab-1=a19, cac-1=a-1, cbc-1=b5 >
Subgroups: 684 in 180 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, M4(2), SD16, SD16, Q16, C2×Q8, C4○D4, C3⋊S3, C3×C6, C3×C6, C3⋊C8, C24, Dic6, C4×S3, C2×Dic3, C3⋊D4, C3×D4, C3×Q8, C8.C22, C3⋊Dic3, C3⋊Dic3, C3×C12, C3×C12, C2×C3⋊S3, C62, C8⋊S3, Dic12, D4.S3, C3⋊Q16, C3×SD16, D4⋊2S3, S3×Q8, C32⋊4C8, C3×C24, C32⋊4Q8, C32⋊4Q8, C4×C3⋊S3, C4×C3⋊S3, C2×C3⋊Dic3, C32⋊7D4, D4×C32, Q8×C32, D4.D6, C24⋊S3, C32⋊5Q16, C32⋊9SD16, C32⋊7Q16, C32×SD16, C12.D6, Q8×C3⋊S3, C24.32D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C3⋊S3, C22×S3, C8.C22, C2×C3⋊S3, S3×D4, C22×C3⋊S3, D4.D6, D4×C3⋊S3, C24.32D6
►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 49 86 144 35 105 13 61 74 132 47 117)(2 68 87 139 36 100 14 56 75 127 48 112)(3 63 88 134 37 119 15 51 76 122 25 107)(4 58 89 129 38 114 16 70 77 141 26 102)(5 53 90 124 39 109 17 65 78 136 27 97)(6 72 91 143 40 104 18 60 79 131 28 116)(7 67 92 138 41 99 19 55 80 126 29 111)(8 62 93 133 42 118 20 50 81 121 30 106)(9 57 94 128 43 113 21 69 82 140 31 101)(10 52 95 123 44 108 22 64 83 135 32 120)(11 71 96 142 45 103 23 59 84 130 33 115)(12 66 73 137 46 98 24 54 85 125 34 110)
(1 105 13 117)(2 104 14 116)(3 103 15 115)(4 102 16 114)(5 101 17 113)(6 100 18 112)(7 99 19 111)(8 98 20 110)(9 97 21 109)(10 120 22 108)(11 119 23 107)(12 118 24 106)(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)(73 133 85 121)(74 132 86 144)(75 131 87 143)(76 130 88 142)(77 129 89 141)(78 128 90 140)(79 127 91 139)(80 126 92 138)(81 125 93 137)(82 124 94 136)(83 123 95 135)(84 122 96 134)
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), (1,49,86,144,35,105,13,61,74,132,47,117)(2,68,87,139,36,100,14,56,75,127,48,112)(3,63,88,134,37,119,15,51,76,122,25,107)(4,58,89,129,38,114,16,70,77,141,26,102)(5,53,90,124,39,109,17,65,78,136,27,97)(6,72,91,143,40,104,18,60,79,131,28,116)(7,67,92,138,41,99,19,55,80,126,29,111)(8,62,93,133,42,118,20,50,81,121,30,106)(9,57,94,128,43,113,21,69,82,140,31,101)(10,52,95,123,44,108,22,64,83,135,32,120)(11,71,96,142,45,103,23,59,84,130,33,115)(12,66,73,137,46,98,24,54,85,125,34,110), (1,105,13,117)(2,104,14,116)(3,103,15,115)(4,102,16,114)(5,101,17,113)(6,100,18,112)(7,99,19,111)(8,98,20,110)(9,97,21,109)(10,120,22,108)(11,119,23,107)(12,118,24,106)(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)(73,133,85,121)(74,132,86,144)(75,131,87,143)(76,130,88,142)(77,129,89,141)(78,128,90,140)(79,127,91,139)(80,126,92,138)(81,125,93,137)(82,124,94,136)(83,123,95,135)(84,122,96,134)>;
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), (1,49,86,144,35,105,13,61,74,132,47,117)(2,68,87,139,36,100,14,56,75,127,48,112)(3,63,88,134,37,119,15,51,76,122,25,107)(4,58,89,129,38,114,16,70,77,141,26,102)(5,53,90,124,39,109,17,65,78,136,27,97)(6,72,91,143,40,104,18,60,79,131,28,116)(7,67,92,138,41,99,19,55,80,126,29,111)(8,62,93,133,42,118,20,50,81,121,30,106)(9,57,94,128,43,113,21,69,82,140,31,101)(10,52,95,123,44,108,22,64,83,135,32,120)(11,71,96,142,45,103,23,59,84,130,33,115)(12,66,73,137,46,98,24,54,85,125,34,110), (1,105,13,117)(2,104,14,116)(3,103,15,115)(4,102,16,114)(5,101,17,113)(6,100,18,112)(7,99,19,111)(8,98,20,110)(9,97,21,109)(10,120,22,108)(11,119,23,107)(12,118,24,106)(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)(73,133,85,121)(74,132,86,144)(75,131,87,143)(76,130,88,142)(77,129,89,141)(78,128,90,140)(79,127,91,139)(80,126,92,138)(81,125,93,137)(82,124,94,136)(83,123,95,135)(84,122,96,134) );
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)], [(1,49,86,144,35,105,13,61,74,132,47,117),(2,68,87,139,36,100,14,56,75,127,48,112),(3,63,88,134,37,119,15,51,76,122,25,107),(4,58,89,129,38,114,16,70,77,141,26,102),(5,53,90,124,39,109,17,65,78,136,27,97),(6,72,91,143,40,104,18,60,79,131,28,116),(7,67,92,138,41,99,19,55,80,126,29,111),(8,62,93,133,42,118,20,50,81,121,30,106),(9,57,94,128,43,113,21,69,82,140,31,101),(10,52,95,123,44,108,22,64,83,135,32,120),(11,71,96,142,45,103,23,59,84,130,33,115),(12,66,73,137,46,98,24,54,85,125,34,110)], [(1,105,13,117),(2,104,14,116),(3,103,15,115),(4,102,16,114),(5,101,17,113),(6,100,18,112),(7,99,19,111),(8,98,20,110),(9,97,21,109),(10,120,22,108),(11,119,23,107),(12,118,24,106),(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),(73,133,85,121),(74,132,86,144),(75,131,87,143),(76,130,88,142),(77,129,89,141),(78,128,90,140),(79,127,91,139),(80,126,92,138),(81,125,93,137),(82,124,94,136),(83,123,95,135),(84,122,96,134)]])
39 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 | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 24A | ··· | 24H |
| 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 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 4 | 18 | 2 | 2 | 2 | 2 | 2 | 4 | 18 | 36 | 36 | 2 | 2 | 2 | 2 | 8 | 8 | 8 | 8 | 4 | 36 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | D6 | C8.C22 | S3×D4 | D4.D6 |
| kernel | C24.32D6 | C24⋊S3 | C32⋊5Q16 | C32⋊9SD16 | C32⋊7Q16 | C32×SD16 | C12.D6 | Q8×C3⋊S3 | 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 | 1 | 4 | 8 |
Matrix representation of C24.32D6 ►in GL6(𝔽73)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 34 | 39 | 39 | 34 |
| 0 | 0 | 34 | 68 | 39 | 5 |
| 0 | 0 | 34 | 39 | 34 | 39 |
| 0 | 0 | 34 | 68 | 34 | 68 |
| 72 | 72 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 65 | 0 | 64 | 0 |
| 0 | 0 | 0 | 65 | 0 | 64 |
| 0 | 0 | 64 | 0 | 8 | 0 |
| 0 | 0 | 0 | 64 | 0 | 8 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 72 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 65 | 0 | 64 | 0 |
| 0 | 0 | 8 | 8 | 9 | 9 |
| 0 | 0 | 64 | 0 | 8 | 0 |
| 0 | 0 | 9 | 9 | 65 | 65 |
G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,34,34,34,34,0,0,39,68,39,68,0,0,39,39,34,34,0,0,34,5,39,68],[72,1,0,0,0,0,72,0,0,0,0,0,0,0,65,0,64,0,0,0,0,65,0,64,0,0,64,0,8,0,0,0,0,64,0,8],[1,72,0,0,0,0,0,72,0,0,0,0,0,0,65,8,64,9,0,0,0,8,0,9,0,0,64,9,8,65,0,0,0,9,0,65] >;
C24.32D6 in GAP, Magma, Sage, TeX
C_{24}._{32}D_6 % in TeX
G:=Group("C24.32D6"); // GroupNames label
G:=SmallGroup(288,772);
// by ID
G=gap.SmallGroup(288,772);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,120,422,135,346,185,80,2693,9414]);
// Polycyclic
G:=Group<a,b,c|a^24=1,b^6=c^2=a^12,b*a*b^-1=a^19,c*a*c^-1=a^-1,c*b*c^-1=b^5>;
// generators/relations