metabelian, supersoluble, monomial
Aliases: C24.94D6, C62.9C8, C24.12Dic3, C32⋊10M5(2), C12.3(C3⋊C8), (C3×C12).7C8, (C3×C24).12C4, (C6×C12).24C4, (C6×C24).21C2, (C2×C24).31S3, C4.(C32⋊4C8), C24.S3⋊9C2, C8.2(C3⋊Dic3), C3⋊2(C12.C8), (C3×C24).71C22, (C2×C12).19Dic3, C12.57(C2×Dic3), C22.(C32⋊4C8), C6.15(C2×C3⋊C8), C8.21(C2×C3⋊S3), (C2×C6).8(C3⋊C8), (C2×C8).7(C3⋊S3), (C3×C6).45(C2×C8), C4.10(C2×C3⋊Dic3), C2.4(C2×C32⋊4C8), (C3×C12).133(C2×C4), (C2×C4).5(C3⋊Dic3), SmallGroup(288,287)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24.94D6
G = < a,b,c | a24=b6=1, c2=a21, ab=ba, cac-1=a17, cbc-1=a12b-1 >
Subgroups: 132 in 78 conjugacy classes, 57 normal (19 characteristic)
C1, C2, C2, C3, C4, C22, C6, C6, C8, C2×C4, C32, C12, C2×C6, C16, C2×C8, C3×C6, C3×C6, C24, C2×C12, M5(2), C3×C12, C62, C3⋊C16, C2×C24, C3×C24, C6×C12, C12.C8, C24.S3, C6×C24, C24.94D6
Quotients: C1, C2, C4, C22, S3, C8, C2×C4, Dic3, D6, C2×C8, C3⋊S3, C3⋊C8, C2×Dic3, M5(2), C3⋊Dic3, C2×C3⋊S3, C2×C3⋊C8, C32⋊4C8, C2×C3⋊Dic3, C12.C8, C2×C32⋊4C8, C24.94D6
►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)
(1 141 100)(2 142 101)(3 143 102)(4 144 103)(5 121 104)(6 122 105)(7 123 106)(8 124 107)(9 125 108)(10 126 109)(11 127 110)(12 128 111)(13 129 112)(14 130 113)(15 131 114)(16 132 115)(17 133 116)(18 134 117)(19 135 118)(20 136 119)(21 137 120)(22 138 97)(23 139 98)(24 140 99)(25 78 62 37 90 50)(26 79 63 38 91 51)(27 80 64 39 92 52)(28 81 65 40 93 53)(29 82 66 41 94 54)(30 83 67 42 95 55)(31 84 68 43 96 56)(32 85 69 44 73 57)(33 86 70 45 74 58)(34 87 71 46 75 59)(35 88 72 47 76 60)(36 89 49 48 77 61)
(1 28 22 25 19 46 16 43 13 40 10 37 7 34 4 31)(2 45 23 42 20 39 17 36 14 33 11 30 8 27 5 48)(3 38 24 35 21 32 18 29 15 26 12 47 9 44 6 41)(49 130 70 127 67 124 64 121 61 142 58 139 55 136 52 133)(50 123 71 144 68 141 65 138 62 135 59 132 56 129 53 126)(51 140 72 137 69 134 66 131 63 128 60 125 57 122 54 143)(73 117 94 114 91 111 88 108 85 105 82 102 79 99 76 120)(74 110 95 107 92 104 89 101 86 98 83 119 80 116 77 113)(75 103 96 100 93 97 90 118 87 115 84 112 81 109 78 106)
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), (1,141,100)(2,142,101)(3,143,102)(4,144,103)(5,121,104)(6,122,105)(7,123,106)(8,124,107)(9,125,108)(10,126,109)(11,127,110)(12,128,111)(13,129,112)(14,130,113)(15,131,114)(16,132,115)(17,133,116)(18,134,117)(19,135,118)(20,136,119)(21,137,120)(22,138,97)(23,139,98)(24,140,99)(25,78,62,37,90,50)(26,79,63,38,91,51)(27,80,64,39,92,52)(28,81,65,40,93,53)(29,82,66,41,94,54)(30,83,67,42,95,55)(31,84,68,43,96,56)(32,85,69,44,73,57)(33,86,70,45,74,58)(34,87,71,46,75,59)(35,88,72,47,76,60)(36,89,49,48,77,61), (1,28,22,25,19,46,16,43,13,40,10,37,7,34,4,31)(2,45,23,42,20,39,17,36,14,33,11,30,8,27,5,48)(3,38,24,35,21,32,18,29,15,26,12,47,9,44,6,41)(49,130,70,127,67,124,64,121,61,142,58,139,55,136,52,133)(50,123,71,144,68,141,65,138,62,135,59,132,56,129,53,126)(51,140,72,137,69,134,66,131,63,128,60,125,57,122,54,143)(73,117,94,114,91,111,88,108,85,105,82,102,79,99,76,120)(74,110,95,107,92,104,89,101,86,98,83,119,80,116,77,113)(75,103,96,100,93,97,90,118,87,115,84,112,81,109,78,106)>;
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), (1,141,100)(2,142,101)(3,143,102)(4,144,103)(5,121,104)(6,122,105)(7,123,106)(8,124,107)(9,125,108)(10,126,109)(11,127,110)(12,128,111)(13,129,112)(14,130,113)(15,131,114)(16,132,115)(17,133,116)(18,134,117)(19,135,118)(20,136,119)(21,137,120)(22,138,97)(23,139,98)(24,140,99)(25,78,62,37,90,50)(26,79,63,38,91,51)(27,80,64,39,92,52)(28,81,65,40,93,53)(29,82,66,41,94,54)(30,83,67,42,95,55)(31,84,68,43,96,56)(32,85,69,44,73,57)(33,86,70,45,74,58)(34,87,71,46,75,59)(35,88,72,47,76,60)(36,89,49,48,77,61), (1,28,22,25,19,46,16,43,13,40,10,37,7,34,4,31)(2,45,23,42,20,39,17,36,14,33,11,30,8,27,5,48)(3,38,24,35,21,32,18,29,15,26,12,47,9,44,6,41)(49,130,70,127,67,124,64,121,61,142,58,139,55,136,52,133)(50,123,71,144,68,141,65,138,62,135,59,132,56,129,53,126)(51,140,72,137,69,134,66,131,63,128,60,125,57,122,54,143)(73,117,94,114,91,111,88,108,85,105,82,102,79,99,76,120)(74,110,95,107,92,104,89,101,86,98,83,119,80,116,77,113)(75,103,96,100,93,97,90,118,87,115,84,112,81,109,78,106) );
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)], [(1,141,100),(2,142,101),(3,143,102),(4,144,103),(5,121,104),(6,122,105),(7,123,106),(8,124,107),(9,125,108),(10,126,109),(11,127,110),(12,128,111),(13,129,112),(14,130,113),(15,131,114),(16,132,115),(17,133,116),(18,134,117),(19,135,118),(20,136,119),(21,137,120),(22,138,97),(23,139,98),(24,140,99),(25,78,62,37,90,50),(26,79,63,38,91,51),(27,80,64,39,92,52),(28,81,65,40,93,53),(29,82,66,41,94,54),(30,83,67,42,95,55),(31,84,68,43,96,56),(32,85,69,44,73,57),(33,86,70,45,74,58),(34,87,71,46,75,59),(35,88,72,47,76,60),(36,89,49,48,77,61)], [(1,28,22,25,19,46,16,43,13,40,10,37,7,34,4,31),(2,45,23,42,20,39,17,36,14,33,11,30,8,27,5,48),(3,38,24,35,21,32,18,29,15,26,12,47,9,44,6,41),(49,130,70,127,67,124,64,121,61,142,58,139,55,136,52,133),(50,123,71,144,68,141,65,138,62,135,59,132,56,129,53,126),(51,140,72,137,69,134,66,131,63,128,60,125,57,122,54,143),(73,117,94,114,91,111,88,108,85,105,82,102,79,99,76,120),(74,110,95,107,92,104,89,101,86,98,83,119,80,116,77,113),(75,103,96,100,93,97,90,118,87,115,84,112,81,109,78,106)]])
84 conjugacy classes
| class | 1 | 2A | 2B | 3A | 3B | 3C | 3D | 4A | 4B | 4C | 6A | ··· | 6L | 8A | 8B | 8C | 8D | 8E | 8F | 12A | ··· | 12P | 16A | ··· | 16H | 24A | ··· | 24AF |
| order | 1 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | ··· | 12 | 16 | ··· | 16 | 24 | ··· | 24 |
| size | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | ··· | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | ··· | 2 | 18 | ··· | 18 | 2 | ··· | 2 |
84 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | - | ||||||||
| image | C1 | C2 | C2 | C4 | C4 | C8 | C8 | S3 | Dic3 | D6 | Dic3 | C3⋊C8 | C3⋊C8 | M5(2) | C12.C8 |
| kernel | C24.94D6 | C24.S3 | C6×C24 | C3×C24 | C6×C12 | C3×C12 | C62 | C2×C24 | C24 | C24 | C2×C12 | C12 | C2×C6 | C32 | C3 |
| # reps | 1 | 2 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 4 | 32 |
Matrix representation of C24.94D6 ►in GL4(𝔽97) generated by
| 22 | 0 | 0 | 0 |
| 0 | 22 | 0 | 0 |
| 0 | 0 | 73 | 0 |
| 0 | 0 | 0 | 88 |
| 35 | 0 | 0 | 0 |
| 2 | 61 | 0 | 0 |
| 0 | 0 | 61 | 0 |
| 0 | 0 | 0 | 62 |
| 77 | 31 | 0 | 0 |
| 41 | 20 | 0 | 0 |
| 0 | 0 | 0 | 35 |
| 0 | 0 | 24 | 0 |
G:=sub<GL(4,GF(97))| [22,0,0,0,0,22,0,0,0,0,73,0,0,0,0,88],[35,2,0,0,0,61,0,0,0,0,61,0,0,0,0,62],[77,41,0,0,31,20,0,0,0,0,0,24,0,0,35,0] >;
C24.94D6 in GAP, Magma, Sage, TeX
C_{24}._{94}D_6 % in TeX
G:=Group("C24.94D6"); // GroupNames label
G:=SmallGroup(288,287);
// by ID
G=gap.SmallGroup(288,287);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,28,253,58,80,2693,9414]);
// Polycyclic
G:=Group<a,b,c|a^24=b^6=1,c^2=a^21,a*b=b*a,c*a*c^-1=a^17,c*b*c^-1=a^12*b^-1>;
// generators/relations