metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C24⋊1C8, C12.34D8, C4.16D24, C12.17Q16, C4.9Dic12, C42.248D6, C12.23M4(2), C8⋊1(C3⋊C8), (C4×C8).6S3, C3⋊1(C8⋊1C8), C6.2(C4⋊C8), (C4×C24).8C2, (C2×C24).9C4, C12.36(C2×C8), (C2×C12).46Q8, C12⋊C8.3C2, (C2×C8).8Dic3, C6.6(C2.D8), (C2×C4).160D12, (C2×C12).398D4, (C2×C4).39Dic6, C2.4(C12⋊C8), C2.1(C24⋊1C4), C6.2(C8.C4), C4.5(C4.Dic3), C2.2(C24.C4), (C4×C12).319C22, C22.16(C4⋊Dic3), C4.7(C2×C3⋊C8), (C2×C6).30(C4⋊C4), (C2×C12).292(C2×C4), (C2×C4).68(C2×Dic3), SmallGroup(192,17)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24⋊1C8
G = < a,b | a24=b8=1, bab-1=a-1 >
Smallest permutation representation of C24⋊1C8
►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)
(1 187 91 151 135 118 53 39)(2 186 92 150 136 117 54 38)(3 185 93 149 137 116 55 37)(4 184 94 148 138 115 56 36)(5 183 95 147 139 114 57 35)(6 182 96 146 140 113 58 34)(7 181 73 145 141 112 59 33)(8 180 74 168 142 111 60 32)(9 179 75 167 143 110 61 31)(10 178 76 166 144 109 62 30)(11 177 77 165 121 108 63 29)(12 176 78 164 122 107 64 28)(13 175 79 163 123 106 65 27)(14 174 80 162 124 105 66 26)(15 173 81 161 125 104 67 25)(16 172 82 160 126 103 68 48)(17 171 83 159 127 102 69 47)(18 170 84 158 128 101 70 46)(19 169 85 157 129 100 71 45)(20 192 86 156 130 99 72 44)(21 191 87 155 131 98 49 43)(22 190 88 154 132 97 50 42)(23 189 89 153 133 120 51 41)(24 188 90 152 134 119 52 40)
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), (1,187,91,151,135,118,53,39)(2,186,92,150,136,117,54,38)(3,185,93,149,137,116,55,37)(4,184,94,148,138,115,56,36)(5,183,95,147,139,114,57,35)(6,182,96,146,140,113,58,34)(7,181,73,145,141,112,59,33)(8,180,74,168,142,111,60,32)(9,179,75,167,143,110,61,31)(10,178,76,166,144,109,62,30)(11,177,77,165,121,108,63,29)(12,176,78,164,122,107,64,28)(13,175,79,163,123,106,65,27)(14,174,80,162,124,105,66,26)(15,173,81,161,125,104,67,25)(16,172,82,160,126,103,68,48)(17,171,83,159,127,102,69,47)(18,170,84,158,128,101,70,46)(19,169,85,157,129,100,71,45)(20,192,86,156,130,99,72,44)(21,191,87,155,131,98,49,43)(22,190,88,154,132,97,50,42)(23,189,89,153,133,120,51,41)(24,188,90,152,134,119,52,40)>;
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), (1,187,91,151,135,118,53,39)(2,186,92,150,136,117,54,38)(3,185,93,149,137,116,55,37)(4,184,94,148,138,115,56,36)(5,183,95,147,139,114,57,35)(6,182,96,146,140,113,58,34)(7,181,73,145,141,112,59,33)(8,180,74,168,142,111,60,32)(9,179,75,167,143,110,61,31)(10,178,76,166,144,109,62,30)(11,177,77,165,121,108,63,29)(12,176,78,164,122,107,64,28)(13,175,79,163,123,106,65,27)(14,174,80,162,124,105,66,26)(15,173,81,161,125,104,67,25)(16,172,82,160,126,103,68,48)(17,171,83,159,127,102,69,47)(18,170,84,158,128,101,70,46)(19,169,85,157,129,100,71,45)(20,192,86,156,130,99,72,44)(21,191,87,155,131,98,49,43)(22,190,88,154,132,97,50,42)(23,189,89,153,133,120,51,41)(24,188,90,152,134,119,52,40) );
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)], [(1,187,91,151,135,118,53,39),(2,186,92,150,136,117,54,38),(3,185,93,149,137,116,55,37),(4,184,94,148,138,115,56,36),(5,183,95,147,139,114,57,35),(6,182,96,146,140,113,58,34),(7,181,73,145,141,112,59,33),(8,180,74,168,142,111,60,32),(9,179,75,167,143,110,61,31),(10,178,76,166,144,109,62,30),(11,177,77,165,121,108,63,29),(12,176,78,164,122,107,64,28),(13,175,79,163,123,106,65,27),(14,174,80,162,124,105,66,26),(15,173,81,161,125,104,67,25),(16,172,82,160,126,103,68,48),(17,171,83,159,127,102,69,47),(18,170,84,158,128,101,70,46),(19,169,85,157,129,100,71,45),(20,192,86,156,130,99,72,44),(21,191,87,155,131,98,49,43),(22,190,88,154,132,97,50,42),(23,189,89,153,133,120,51,41),(24,188,90,152,134,119,52,40)]])
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | 6B | 6C | 8A | ··· | 8H | 8I | ··· | 8P | 12A | ··· | 12L | 24A | ··· | 24P |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 8 | ··· | 8 | 8 | ··· | 8 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 12 | ··· | 12 | 2 | ··· | 2 | 2 | ··· | 2 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | - | + | - | - | + | + | - | |||||||
| image | C1 | C2 | C2 | C4 | C8 | S3 | D4 | Q8 | D6 | Dic3 | M4(2) | D8 | Q16 | C3⋊C8 | Dic6 | D12 | C8.C4 | D24 | Dic12 | C4.Dic3 | C24.C4 |
| kernel | C24⋊1C8 | C12⋊C8 | C4×C24 | C2×C24 | C24 | C4×C8 | C2×C12 | C2×C12 | C42 | C2×C8 | C12 | C12 | C12 | C8 | C2×C4 | C2×C4 | C6 | C4 | C4 | C4 | C2 |
| # reps | 1 | 2 | 1 | 4 | 8 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 2 | 2 | 4 | 4 | 4 | 4 | 8 |
Matrix representation of C24⋊1C8 ►in GL3(𝔽73) generated by
| 72 | 0 | 0 |
| 0 | 52 | 0 |
| 0 | 0 | 66 |
| 63 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 27 | 0 |
G:=sub<GL(3,GF(73))| [72,0,0,0,52,0,0,0,66],[63,0,0,0,0,27,0,1,0] >;
C24⋊1C8 in GAP, Magma, Sage, TeX
C_{24}\rtimes_1C_8 % in TeX
G:=Group("C24:1C8"); // GroupNames label
G:=SmallGroup(192,17);
// by ID
G=gap.SmallGroup(192,17);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,28,141,288,100,1123,136,6278]);
// Polycyclic
G:=Group<a,b|a^24=b^8=1,b*a*b^-1=a^-1>;
// generators/relations
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