metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C24⋊5C8, C42.280D6, C12.15M4(2), C8⋊3(C3⋊C8), C6.2(C4×C8), C3⋊2(C8⋊C8), (C4×C8).14S3, (C2×C24).22C4, (C4×C24).23C2, C12.39(C2×C8), C6.4(C8⋊C4), (C2×C6).15C42, (C2×C8).12Dic3, C2.2(C24⋊C4), C4.13(C8⋊S3), (C4×C12).336C22, C22.15(C4×Dic3), C2.3(C4×C3⋊C8), C4.11(C2×C3⋊C8), (C2×C3⋊C8).11C4, (C4×C3⋊C8).17C2, (C2×C4).165(C4×S3), (C2×C12).239(C2×C4), (C2×C4).89(C2×Dic3), SmallGroup(192,14)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24⋊C8
G = < a,b | a24=b8=1, bab-1=a5 >
Subgroups: 104 in 66 conjugacy classes, 51 normal (13 characteristic)
C1, C2, C2, C3, C4, C22, C6, C6, C8, C8, C2×C4, C2×C4, C12, C2×C6, C42, C2×C8, C2×C8, C3⋊C8, C24, C2×C12, C2×C12, C4×C8, C4×C8, C2×C3⋊C8, C4×C12, C2×C24, C8⋊C8, C4×C3⋊C8, C4×C24, C24⋊C8
Quotients: C1, C2, C4, C22, S3, C8, C2×C4, Dic3, D6, C42, C2×C8, M4(2), C3⋊C8, C4×S3, C2×Dic3, C4×C8, C8⋊C4, C8⋊S3, C2×C3⋊C8, C4×Dic3, C8⋊C8, C4×C3⋊C8, C24⋊C4, C24⋊C8
►Regular action on 192 points
Generators in S192
(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)(145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168)(169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192)
(1 139 32 64 166 89 102 184)(2 144 33 69 167 94 103 189)(3 125 34 50 168 75 104 170)(4 130 35 55 145 80 105 175)(5 135 36 60 146 85 106 180)(6 140 37 65 147 90 107 185)(7 121 38 70 148 95 108 190)(8 126 39 51 149 76 109 171)(9 131 40 56 150 81 110 176)(10 136 41 61 151 86 111 181)(11 141 42 66 152 91 112 186)(12 122 43 71 153 96 113 191)(13 127 44 52 154 77 114 172)(14 132 45 57 155 82 115 177)(15 137 46 62 156 87 116 182)(16 142 47 67 157 92 117 187)(17 123 48 72 158 73 118 192)(18 128 25 53 159 78 119 173)(19 133 26 58 160 83 120 178)(20 138 27 63 161 88 97 183)(21 143 28 68 162 93 98 188)(22 124 29 49 163 74 99 169)(23 129 30 54 164 79 100 174)(24 134 31 59 165 84 101 179)
G:=sub<Sym(192)| (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)(145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168)(169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192), (1,139,32,64,166,89,102,184)(2,144,33,69,167,94,103,189)(3,125,34,50,168,75,104,170)(4,130,35,55,145,80,105,175)(5,135,36,60,146,85,106,180)(6,140,37,65,147,90,107,185)(7,121,38,70,148,95,108,190)(8,126,39,51,149,76,109,171)(9,131,40,56,150,81,110,176)(10,136,41,61,151,86,111,181)(11,141,42,66,152,91,112,186)(12,122,43,71,153,96,113,191)(13,127,44,52,154,77,114,172)(14,132,45,57,155,82,115,177)(15,137,46,62,156,87,116,182)(16,142,47,67,157,92,117,187)(17,123,48,72,158,73,118,192)(18,128,25,53,159,78,119,173)(19,133,26,58,160,83,120,178)(20,138,27,63,161,88,97,183)(21,143,28,68,162,93,98,188)(22,124,29,49,163,74,99,169)(23,129,30,54,164,79,100,174)(24,134,31,59,165,84,101,179)>;
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)(145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168)(169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192), (1,139,32,64,166,89,102,184)(2,144,33,69,167,94,103,189)(3,125,34,50,168,75,104,170)(4,130,35,55,145,80,105,175)(5,135,36,60,146,85,106,180)(6,140,37,65,147,90,107,185)(7,121,38,70,148,95,108,190)(8,126,39,51,149,76,109,171)(9,131,40,56,150,81,110,176)(10,136,41,61,151,86,111,181)(11,141,42,66,152,91,112,186)(12,122,43,71,153,96,113,191)(13,127,44,52,154,77,114,172)(14,132,45,57,155,82,115,177)(15,137,46,62,156,87,116,182)(16,142,47,67,157,92,117,187)(17,123,48,72,158,73,118,192)(18,128,25,53,159,78,119,173)(19,133,26,58,160,83,120,178)(20,138,27,63,161,88,97,183)(21,143,28,68,162,93,98,188)(22,124,29,49,163,74,99,169)(23,129,30,54,164,79,100,174)(24,134,31,59,165,84,101,179) );
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),(145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168),(169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192)], [(1,139,32,64,166,89,102,184),(2,144,33,69,167,94,103,189),(3,125,34,50,168,75,104,170),(4,130,35,55,145,80,105,175),(5,135,36,60,146,85,106,180),(6,140,37,65,147,90,107,185),(7,121,38,70,148,95,108,190),(8,126,39,51,149,76,109,171),(9,131,40,56,150,81,110,176),(10,136,41,61,151,86,111,181),(11,141,42,66,152,91,112,186),(12,122,43,71,153,96,113,191),(13,127,44,52,154,77,114,172),(14,132,45,57,155,82,115,177),(15,137,46,62,156,87,116,182),(16,142,47,67,157,92,117,187),(17,123,48,72,158,73,118,192),(18,128,25,53,159,78,119,173),(19,133,26,58,160,83,120,178),(20,138,27,63,161,88,97,183),(21,143,28,68,162,93,98,188),(22,124,29,49,163,74,99,169),(23,129,30,54,164,79,100,174),(24,134,31,59,165,84,101,179)]])
72 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | ··· | 4L | 6A | 6B | 6C | 8A | ··· | 8H | 8I | ··· | 8X | 12A | ··· | 12L | 24A | ··· | 24P |
| order | 1 | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 6 | 6 | 6 | 8 | ··· | 8 | 8 | ··· | 8 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 6 | ··· | 6 | 2 | ··· | 2 | 2 | ··· | 2 |
72 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | |||||||
| image | C1 | C2 | C2 | C4 | C4 | C8 | S3 | D6 | Dic3 | M4(2) | C3⋊C8 | C4×S3 | C8⋊S3 |
| kernel | C24⋊C8 | C4×C3⋊C8 | C4×C24 | C2×C3⋊C8 | C2×C24 | C24 | C4×C8 | C42 | C2×C8 | C12 | C8 | C2×C4 | C4 |
| # reps | 1 | 2 | 1 | 8 | 4 | 16 | 1 | 1 | 2 | 8 | 8 | 4 | 16 |
Matrix representation of C24⋊C8 ►in GL4(𝔽73) generated by
| 46 | 46 | 0 | 0 |
| 27 | 0 | 0 | 0 |
| 0 | 0 | 3 | 70 |
| 0 | 0 | 3 | 6 |
| 22 | 22 | 0 | 0 |
| 0 | 51 | 0 | 0 |
| 0 | 0 | 66 | 7 |
| 0 | 0 | 14 | 7 |
G:=sub<GL(4,GF(73))| [46,27,0,0,46,0,0,0,0,0,3,3,0,0,70,6],[22,0,0,0,22,51,0,0,0,0,66,14,0,0,7,7] >;
C24⋊C8 in GAP, Magma, Sage, TeX
C_{24}\rtimes C_8 % in TeX
G:=Group("C24:C8"); // GroupNames label
G:=SmallGroup(192,14);
// by ID
G=gap.SmallGroup(192,14);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,28,477,64,100,136,6278]);
// Polycyclic
G:=Group<a,b|a^24=b^8=1,b*a*b^-1=a^5>;
// generators/relations