metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C24⋊2C8, C12.28SD16, C42.247D6, C12.22M4(2), C8⋊2(C3⋊C8), C3⋊1(C8⋊2C8), C6.1(C4⋊C8), (C2×C24).8C4, (C4×C8).12S3, (C4×C24).14C2, C12.35(C2×C8), (C2×C12).45Q8, C12⋊C8.2C2, C6.5(C4.Q8), (C2×C4).159D12, (C2×C12).397D4, (C2×C4).38Dic6, (C2×C8).10Dic3, C2.1(C8⋊Dic3), C2.3(C12⋊C8), C4.15(C24⋊C2), C6.1(C8.C4), C4.4(C4.Dic3), C2.1(C24.C4), (C4×C12).318C22, C22.15(C4⋊Dic3), C4.6(C2×C3⋊C8), (C2×C6).29(C4⋊C4), (C2×C12).291(C2×C4), (C2×C4).67(C2×Dic3), SmallGroup(192,16)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24⋊2C8
G = < a,b | a24=b8=1, bab-1=a11 >
Smallest permutation representation of C24⋊2C8
►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)
(1 122 154 86 28 64 169 117)(2 133 155 73 29 51 170 104)(3 144 156 84 30 62 171 115)(4 131 157 95 31 49 172 102)(5 142 158 82 32 60 173 113)(6 129 159 93 33 71 174 100)(7 140 160 80 34 58 175 111)(8 127 161 91 35 69 176 98)(9 138 162 78 36 56 177 109)(10 125 163 89 37 67 178 120)(11 136 164 76 38 54 179 107)(12 123 165 87 39 65 180 118)(13 134 166 74 40 52 181 105)(14 121 167 85 41 63 182 116)(15 132 168 96 42 50 183 103)(16 143 145 83 43 61 184 114)(17 130 146 94 44 72 185 101)(18 141 147 81 45 59 186 112)(19 128 148 92 46 70 187 99)(20 139 149 79 47 57 188 110)(21 126 150 90 48 68 189 97)(22 137 151 77 25 55 190 108)(23 124 152 88 26 66 191 119)(24 135 153 75 27 53 192 106)
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), (1,122,154,86,28,64,169,117)(2,133,155,73,29,51,170,104)(3,144,156,84,30,62,171,115)(4,131,157,95,31,49,172,102)(5,142,158,82,32,60,173,113)(6,129,159,93,33,71,174,100)(7,140,160,80,34,58,175,111)(8,127,161,91,35,69,176,98)(9,138,162,78,36,56,177,109)(10,125,163,89,37,67,178,120)(11,136,164,76,38,54,179,107)(12,123,165,87,39,65,180,118)(13,134,166,74,40,52,181,105)(14,121,167,85,41,63,182,116)(15,132,168,96,42,50,183,103)(16,143,145,83,43,61,184,114)(17,130,146,94,44,72,185,101)(18,141,147,81,45,59,186,112)(19,128,148,92,46,70,187,99)(20,139,149,79,47,57,188,110)(21,126,150,90,48,68,189,97)(22,137,151,77,25,55,190,108)(23,124,152,88,26,66,191,119)(24,135,153,75,27,53,192,106)>;
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)(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), (1,122,154,86,28,64,169,117)(2,133,155,73,29,51,170,104)(3,144,156,84,30,62,171,115)(4,131,157,95,31,49,172,102)(5,142,158,82,32,60,173,113)(6,129,159,93,33,71,174,100)(7,140,160,80,34,58,175,111)(8,127,161,91,35,69,176,98)(9,138,162,78,36,56,177,109)(10,125,163,89,37,67,178,120)(11,136,164,76,38,54,179,107)(12,123,165,87,39,65,180,118)(13,134,166,74,40,52,181,105)(14,121,167,85,41,63,182,116)(15,132,168,96,42,50,183,103)(16,143,145,83,43,61,184,114)(17,130,146,94,44,72,185,101)(18,141,147,81,45,59,186,112)(19,128,148,92,46,70,187,99)(20,139,149,79,47,57,188,110)(21,126,150,90,48,68,189,97)(22,137,151,77,25,55,190,108)(23,124,152,88,26,66,191,119)(24,135,153,75,27,53,192,106) );
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),(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)], [(1,122,154,86,28,64,169,117),(2,133,155,73,29,51,170,104),(3,144,156,84,30,62,171,115),(4,131,157,95,31,49,172,102),(5,142,158,82,32,60,173,113),(6,129,159,93,33,71,174,100),(7,140,160,80,34,58,175,111),(8,127,161,91,35,69,176,98),(9,138,162,78,36,56,177,109),(10,125,163,89,37,67,178,120),(11,136,164,76,38,54,179,107),(12,123,165,87,39,65,180,118),(13,134,166,74,40,52,181,105),(14,121,167,85,41,63,182,116),(15,132,168,96,42,50,183,103),(16,143,145,83,43,61,184,114),(17,130,146,94,44,72,185,101),(18,141,147,81,45,59,186,112),(19,128,148,92,46,70,187,99),(20,139,149,79,47,57,188,110),(21,126,150,90,48,68,189,97),(22,137,151,77,25,55,190,108),(23,124,152,88,26,66,191,119),(24,135,153,75,27,53,192,106)]])
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 |
| type | + | + | + | + | + | - | + | - | - | + | |||||||||
| image | C1 | C2 | C2 | C4 | C8 | S3 | D4 | Q8 | D6 | Dic3 | M4(2) | SD16 | C3⋊C8 | Dic6 | D12 | C8.C4 | C24⋊C2 | C4.Dic3 | C24.C4 |
| kernel | C24⋊2C8 | C12⋊C8 | C4×C24 | C2×C24 | C24 | C4×C8 | C2×C12 | C2×C12 | C42 | C2×C8 | C12 | C12 | C8 | C2×C4 | C2×C4 | C6 | C4 | C4 | C2 |
| # reps | 1 | 2 | 1 | 4 | 8 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 2 | 2 | 4 | 8 | 4 | 8 |
Matrix representation of C24⋊2C8 ►in GL4(𝔽73) generated by
| 14 | 66 | 0 | 0 |
| 7 | 7 | 0 | 0 |
| 0 | 0 | 36 | 25 |
| 0 | 0 | 48 | 11 |
| 43 | 12 | 0 | 0 |
| 55 | 30 | 0 | 0 |
| 0 | 0 | 46 | 72 |
| 0 | 0 | 26 | 27 |
G:=sub<GL(4,GF(73))| [14,7,0,0,66,7,0,0,0,0,36,48,0,0,25,11],[43,55,0,0,12,30,0,0,0,0,46,26,0,0,72,27] >;
C24⋊2C8 in GAP, Magma, Sage, TeX
C_{24}\rtimes_2C_8 % in TeX
G:=Group("C24:2C8"); // GroupNames label
G:=SmallGroup(192,16);
// by ID
G=gap.SmallGroup(192,16);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,28,141,64,100,1123,136,6278]);
// Polycyclic
G:=Group<a,b|a^24=b^8=1,b*a*b^-1=a^11>;
// generators/relations
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