direct product, abelian, monomial, 2-elementary
Aliases: C8×C24, SmallGroup(192,127)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C8×C24 |
| C1 — C8×C24 |
| C1 — C8×C24 |
Generators and relations for C8×C24
G = < a,b | a8=b24=1, ab=ba >
Subgroups: 74, all normal (8 characteristic)
C1, C2, C3, C4, C22, C6, C8, C2×C4, C12, C2×C6, C42, C2×C8, C24, C2×C12, C4×C8, C4×C12, C2×C24, C82, C4×C24, C8×C24
Quotients: C1, C2, C3, C4, C22, C6, C8, C2×C4, C12, C2×C6, C42, C2×C8, C24, C2×C12, C4×C8, C4×C12, C2×C24, C82, C4×C24, C8×C24
►Regular action on 192 points
Generators in S192
(1 168 86 42 102 51 186 137)(2 145 87 43 103 52 187 138)(3 146 88 44 104 53 188 139)(4 147 89 45 105 54 189 140)(5 148 90 46 106 55 190 141)(6 149 91 47 107 56 191 142)(7 150 92 48 108 57 192 143)(8 151 93 25 109 58 169 144)(9 152 94 26 110 59 170 121)(10 153 95 27 111 60 171 122)(11 154 96 28 112 61 172 123)(12 155 73 29 113 62 173 124)(13 156 74 30 114 63 174 125)(14 157 75 31 115 64 175 126)(15 158 76 32 116 65 176 127)(16 159 77 33 117 66 177 128)(17 160 78 34 118 67 178 129)(18 161 79 35 119 68 179 130)(19 162 80 36 120 69 180 131)(20 163 81 37 97 70 181 132)(21 164 82 38 98 71 182 133)(22 165 83 39 99 72 183 134)(23 166 84 40 100 49 184 135)(24 167 85 41 101 50 185 136)
(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)
G:=sub<Sym(192)| (1,168,86,42,102,51,186,137)(2,145,87,43,103,52,187,138)(3,146,88,44,104,53,188,139)(4,147,89,45,105,54,189,140)(5,148,90,46,106,55,190,141)(6,149,91,47,107,56,191,142)(7,150,92,48,108,57,192,143)(8,151,93,25,109,58,169,144)(9,152,94,26,110,59,170,121)(10,153,95,27,111,60,171,122)(11,154,96,28,112,61,172,123)(12,155,73,29,113,62,173,124)(13,156,74,30,114,63,174,125)(14,157,75,31,115,64,175,126)(15,158,76,32,116,65,176,127)(16,159,77,33,117,66,177,128)(17,160,78,34,118,67,178,129)(18,161,79,35,119,68,179,130)(19,162,80,36,120,69,180,131)(20,163,81,37,97,70,181,132)(21,164,82,38,98,71,182,133)(22,165,83,39,99,72,183,134)(23,166,84,40,100,49,184,135)(24,167,85,41,101,50,185,136), (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)>;
G:=Group( (1,168,86,42,102,51,186,137)(2,145,87,43,103,52,187,138)(3,146,88,44,104,53,188,139)(4,147,89,45,105,54,189,140)(5,148,90,46,106,55,190,141)(6,149,91,47,107,56,191,142)(7,150,92,48,108,57,192,143)(8,151,93,25,109,58,169,144)(9,152,94,26,110,59,170,121)(10,153,95,27,111,60,171,122)(11,154,96,28,112,61,172,123)(12,155,73,29,113,62,173,124)(13,156,74,30,114,63,174,125)(14,157,75,31,115,64,175,126)(15,158,76,32,116,65,176,127)(16,159,77,33,117,66,177,128)(17,160,78,34,118,67,178,129)(18,161,79,35,119,68,179,130)(19,162,80,36,120,69,180,131)(20,163,81,37,97,70,181,132)(21,164,82,38,98,71,182,133)(22,165,83,39,99,72,183,134)(23,166,84,40,100,49,184,135)(24,167,85,41,101,50,185,136), (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) );
G=PermutationGroup([[(1,168,86,42,102,51,186,137),(2,145,87,43,103,52,187,138),(3,146,88,44,104,53,188,139),(4,147,89,45,105,54,189,140),(5,148,90,46,106,55,190,141),(6,149,91,47,107,56,191,142),(7,150,92,48,108,57,192,143),(8,151,93,25,109,58,169,144),(9,152,94,26,110,59,170,121),(10,153,95,27,111,60,171,122),(11,154,96,28,112,61,172,123),(12,155,73,29,113,62,173,124),(13,156,74,30,114,63,174,125),(14,157,75,31,115,64,175,126),(15,158,76,32,116,65,176,127),(16,159,77,33,117,66,177,128),(17,160,78,34,118,67,178,129),(18,161,79,35,119,68,179,130),(19,162,80,36,120,69,180,131),(20,163,81,37,97,70,181,132),(21,164,82,38,98,71,182,133),(22,165,83,39,99,72,183,134),(23,166,84,40,100,49,184,135),(24,167,85,41,101,50,185,136)], [(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)]])
192 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | ··· | 4L | 6A | ··· | 6F | 8A | ··· | 8AV | 12A | ··· | 12X | 24A | ··· | 24CR |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | ··· | 4 | 6 | ··· | 6 | 8 | ··· | 8 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
192 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | ||||||
| image | C1 | C2 | C3 | C4 | C6 | C8 | C12 | C24 |
| kernel | C8×C24 | C4×C24 | C82 | C2×C24 | C4×C8 | C24 | C2×C8 | C8 |
| # reps | 1 | 3 | 2 | 12 | 6 | 48 | 24 | 96 |
Matrix representation of C8×C24 ►in GL2(𝔽73) generated by
| 51 | 0 |
| 0 | 63 |
| 70 | 0 |
| 0 | 21 |
G:=sub<GL(2,GF(73))| [51,0,0,63],[70,0,0,21] >;
C8×C24 in GAP, Magma, Sage, TeX
C_8\times C_{24} % in TeX
G:=Group("C8xC24"); // GroupNames label
G:=SmallGroup(192,127);
// by ID
G=gap.SmallGroup(192,127);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-2,84,176,136,172]);
// Polycyclic
G:=Group<a,b|a^8=b^24=1,a*b=b*a>;
// generators/relations