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 [×4], C4 [×2], C22, C6 [×4], C6 [×4], C8 [×2], C2×C4, C32, C12 [×8], C2×C6 [×4], C16 [×2], C2×C8, C3×C6, C3×C6, C24 [×8], C2×C12 [×4], M5(2), C3×C12 [×2], C62, C3⋊C16 [×8], C2×C24 [×4], C3×C24 [×2], C6×C12, C12.C8 [×4], C24.S3 [×2], C6×C24, C24.94D6
Quotients: C1, C2 [×3], C4 [×2], C22, S3 [×4], C8 [×2], C2×C4, Dic3 [×8], D6 [×4], C2×C8, C3⋊S3, C3⋊C8 [×8], C2×Dic3 [×4], M5(2), C3⋊Dic3 [×2], C2×C3⋊S3, C2×C3⋊C8 [×4], C32⋊4C8 [×2], C2×C3⋊Dic3, C12.C8 [×4], 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 45 112)(2 46 113)(3 47 114)(4 48 115)(5 25 116)(6 26 117)(7 27 118)(8 28 119)(9 29 120)(10 30 97)(11 31 98)(12 32 99)(13 33 100)(14 34 101)(15 35 102)(16 36 103)(17 37 104)(18 38 105)(19 39 106)(20 40 107)(21 41 108)(22 42 109)(23 43 110)(24 44 111)(49 122 87 61 134 75)(50 123 88 62 135 76)(51 124 89 63 136 77)(52 125 90 64 137 78)(53 126 91 65 138 79)(54 127 92 66 139 80)(55 128 93 67 140 81)(56 129 94 68 141 82)(57 130 95 69 142 83)(58 131 96 70 143 84)(59 132 73 71 144 85)(60 133 74 72 121 86)
(1 60 22 57 19 54 16 51 13 72 10 69 7 66 4 63)(2 53 23 50 20 71 17 68 14 65 11 62 8 59 5 56)(3 70 24 67 21 64 18 61 15 58 12 55 9 52 6 49)(25 94 46 91 43 88 40 85 37 82 34 79 31 76 28 73)(26 87 47 84 44 81 41 78 38 75 35 96 32 93 29 90)(27 80 48 77 45 74 42 95 39 92 36 89 33 86 30 83)(97 130 118 127 115 124 112 121 109 142 106 139 103 136 100 133)(98 123 119 144 116 141 113 138 110 135 107 132 104 129 101 126)(99 140 120 137 117 134 114 131 111 128 108 125 105 122 102 143)
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), (1,45,112)(2,46,113)(3,47,114)(4,48,115)(5,25,116)(6,26,117)(7,27,118)(8,28,119)(9,29,120)(10,30,97)(11,31,98)(12,32,99)(13,33,100)(14,34,101)(15,35,102)(16,36,103)(17,37,104)(18,38,105)(19,39,106)(20,40,107)(21,41,108)(22,42,109)(23,43,110)(24,44,111)(49,122,87,61,134,75)(50,123,88,62,135,76)(51,124,89,63,136,77)(52,125,90,64,137,78)(53,126,91,65,138,79)(54,127,92,66,139,80)(55,128,93,67,140,81)(56,129,94,68,141,82)(57,130,95,69,142,83)(58,131,96,70,143,84)(59,132,73,71,144,85)(60,133,74,72,121,86), (1,60,22,57,19,54,16,51,13,72,10,69,7,66,4,63)(2,53,23,50,20,71,17,68,14,65,11,62,8,59,5,56)(3,70,24,67,21,64,18,61,15,58,12,55,9,52,6,49)(25,94,46,91,43,88,40,85,37,82,34,79,31,76,28,73)(26,87,47,84,44,81,41,78,38,75,35,96,32,93,29,90)(27,80,48,77,45,74,42,95,39,92,36,89,33,86,30,83)(97,130,118,127,115,124,112,121,109,142,106,139,103,136,100,133)(98,123,119,144,116,141,113,138,110,135,107,132,104,129,101,126)(99,140,120,137,117,134,114,131,111,128,108,125,105,122,102,143)>;
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), (1,45,112)(2,46,113)(3,47,114)(4,48,115)(5,25,116)(6,26,117)(7,27,118)(8,28,119)(9,29,120)(10,30,97)(11,31,98)(12,32,99)(13,33,100)(14,34,101)(15,35,102)(16,36,103)(17,37,104)(18,38,105)(19,39,106)(20,40,107)(21,41,108)(22,42,109)(23,43,110)(24,44,111)(49,122,87,61,134,75)(50,123,88,62,135,76)(51,124,89,63,136,77)(52,125,90,64,137,78)(53,126,91,65,138,79)(54,127,92,66,139,80)(55,128,93,67,140,81)(56,129,94,68,141,82)(57,130,95,69,142,83)(58,131,96,70,143,84)(59,132,73,71,144,85)(60,133,74,72,121,86), (1,60,22,57,19,54,16,51,13,72,10,69,7,66,4,63)(2,53,23,50,20,71,17,68,14,65,11,62,8,59,5,56)(3,70,24,67,21,64,18,61,15,58,12,55,9,52,6,49)(25,94,46,91,43,88,40,85,37,82,34,79,31,76,28,73)(26,87,47,84,44,81,41,78,38,75,35,96,32,93,29,90)(27,80,48,77,45,74,42,95,39,92,36,89,33,86,30,83)(97,130,118,127,115,124,112,121,109,142,106,139,103,136,100,133)(98,123,119,144,116,141,113,138,110,135,107,132,104,129,101,126)(99,140,120,137,117,134,114,131,111,128,108,125,105,122,102,143) );
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)], [(1,45,112),(2,46,113),(3,47,114),(4,48,115),(5,25,116),(6,26,117),(7,27,118),(8,28,119),(9,29,120),(10,30,97),(11,31,98),(12,32,99),(13,33,100),(14,34,101),(15,35,102),(16,36,103),(17,37,104),(18,38,105),(19,39,106),(20,40,107),(21,41,108),(22,42,109),(23,43,110),(24,44,111),(49,122,87,61,134,75),(50,123,88,62,135,76),(51,124,89,63,136,77),(52,125,90,64,137,78),(53,126,91,65,138,79),(54,127,92,66,139,80),(55,128,93,67,140,81),(56,129,94,68,141,82),(57,130,95,69,142,83),(58,131,96,70,143,84),(59,132,73,71,144,85),(60,133,74,72,121,86)], [(1,60,22,57,19,54,16,51,13,72,10,69,7,66,4,63),(2,53,23,50,20,71,17,68,14,65,11,62,8,59,5,56),(3,70,24,67,21,64,18,61,15,58,12,55,9,52,6,49),(25,94,46,91,43,88,40,85,37,82,34,79,31,76,28,73),(26,87,47,84,44,81,41,78,38,75,35,96,32,93,29,90),(27,80,48,77,45,74,42,95,39,92,36,89,33,86,30,83),(97,130,118,127,115,124,112,121,109,142,106,139,103,136,100,133),(98,123,119,144,116,141,113,138,110,135,107,132,104,129,101,126),(99,140,120,137,117,134,114,131,111,128,108,125,105,122,102,143)])
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