direct product, metacyclic, nilpotent (class 3), monomial, 2-elementary
Aliases: D8×C23, D4⋊C46, C8⋊1C46, C184⋊5C2, C46.14D4, C92.17C22, (D4×C23)⋊4C2, C4.1(C2×C46), C2.3(D4×C23), SmallGroup(368,24)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D8×C23
G = < a,b,c | a23=b8=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of D8×C23
►On 184 points
Generators in S184
(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)
(1 184 52 112 158 89 31 132)(2 162 53 113 159 90 32 133)(3 163 54 114 160 91 33 134)(4 164 55 115 161 92 34 135)(5 165 56 93 139 70 35 136)(6 166 57 94 140 71 36 137)(7 167 58 95 141 72 37 138)(8 168 59 96 142 73 38 116)(9 169 60 97 143 74 39 117)(10 170 61 98 144 75 40 118)(11 171 62 99 145 76 41 119)(12 172 63 100 146 77 42 120)(13 173 64 101 147 78 43 121)(14 174 65 102 148 79 44 122)(15 175 66 103 149 80 45 123)(16 176 67 104 150 81 46 124)(17 177 68 105 151 82 24 125)(18 178 69 106 152 83 25 126)(19 179 47 107 153 84 26 127)(20 180 48 108 154 85 27 128)(21 181 49 109 155 86 28 129)(22 182 50 110 156 87 29 130)(23 183 51 111 157 88 30 131)
(1 132)(2 133)(3 134)(4 135)(5 136)(6 137)(7 138)(8 116)(9 117)(10 118)(11 119)(12 120)(13 121)(14 122)(15 123)(16 124)(17 125)(18 126)(19 127)(20 128)(21 129)(22 130)(23 131)(24 177)(25 178)(26 179)(27 180)(28 181)(29 182)(30 183)(31 184)(32 162)(33 163)(34 164)(35 165)(36 166)(37 167)(38 168)(39 169)(40 170)(41 171)(42 172)(43 173)(44 174)(45 175)(46 176)(47 84)(48 85)(49 86)(50 87)(51 88)(52 89)(53 90)(54 91)(55 92)(56 70)(57 71)(58 72)(59 73)(60 74)(61 75)(62 76)(63 77)(64 78)(65 79)(66 80)(67 81)(68 82)(69 83)(93 139)(94 140)(95 141)(96 142)(97 143)(98 144)(99 145)(100 146)(101 147)(102 148)(103 149)(104 150)(105 151)(106 152)(107 153)(108 154)(109 155)(110 156)(111 157)(112 158)(113 159)(114 160)(115 161)
G:=sub<Sym(184)| (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), (1,184,52,112,158,89,31,132)(2,162,53,113,159,90,32,133)(3,163,54,114,160,91,33,134)(4,164,55,115,161,92,34,135)(5,165,56,93,139,70,35,136)(6,166,57,94,140,71,36,137)(7,167,58,95,141,72,37,138)(8,168,59,96,142,73,38,116)(9,169,60,97,143,74,39,117)(10,170,61,98,144,75,40,118)(11,171,62,99,145,76,41,119)(12,172,63,100,146,77,42,120)(13,173,64,101,147,78,43,121)(14,174,65,102,148,79,44,122)(15,175,66,103,149,80,45,123)(16,176,67,104,150,81,46,124)(17,177,68,105,151,82,24,125)(18,178,69,106,152,83,25,126)(19,179,47,107,153,84,26,127)(20,180,48,108,154,85,27,128)(21,181,49,109,155,86,28,129)(22,182,50,110,156,87,29,130)(23,183,51,111,157,88,30,131), (1,132)(2,133)(3,134)(4,135)(5,136)(6,137)(7,138)(8,116)(9,117)(10,118)(11,119)(12,120)(13,121)(14,122)(15,123)(16,124)(17,125)(18,126)(19,127)(20,128)(21,129)(22,130)(23,131)(24,177)(25,178)(26,179)(27,180)(28,181)(29,182)(30,183)(31,184)(32,162)(33,163)(34,164)(35,165)(36,166)(37,167)(38,168)(39,169)(40,170)(41,171)(42,172)(43,173)(44,174)(45,175)(46,176)(47,84)(48,85)(49,86)(50,87)(51,88)(52,89)(53,90)(54,91)(55,92)(56,70)(57,71)(58,72)(59,73)(60,74)(61,75)(62,76)(63,77)(64,78)(65,79)(66,80)(67,81)(68,82)(69,83)(93,139)(94,140)(95,141)(96,142)(97,143)(98,144)(99,145)(100,146)(101,147)(102,148)(103,149)(104,150)(105,151)(106,152)(107,153)(108,154)(109,155)(110,156)(111,157)(112,158)(113,159)(114,160)(115,161)>;
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), (1,184,52,112,158,89,31,132)(2,162,53,113,159,90,32,133)(3,163,54,114,160,91,33,134)(4,164,55,115,161,92,34,135)(5,165,56,93,139,70,35,136)(6,166,57,94,140,71,36,137)(7,167,58,95,141,72,37,138)(8,168,59,96,142,73,38,116)(9,169,60,97,143,74,39,117)(10,170,61,98,144,75,40,118)(11,171,62,99,145,76,41,119)(12,172,63,100,146,77,42,120)(13,173,64,101,147,78,43,121)(14,174,65,102,148,79,44,122)(15,175,66,103,149,80,45,123)(16,176,67,104,150,81,46,124)(17,177,68,105,151,82,24,125)(18,178,69,106,152,83,25,126)(19,179,47,107,153,84,26,127)(20,180,48,108,154,85,27,128)(21,181,49,109,155,86,28,129)(22,182,50,110,156,87,29,130)(23,183,51,111,157,88,30,131), (1,132)(2,133)(3,134)(4,135)(5,136)(6,137)(7,138)(8,116)(9,117)(10,118)(11,119)(12,120)(13,121)(14,122)(15,123)(16,124)(17,125)(18,126)(19,127)(20,128)(21,129)(22,130)(23,131)(24,177)(25,178)(26,179)(27,180)(28,181)(29,182)(30,183)(31,184)(32,162)(33,163)(34,164)(35,165)(36,166)(37,167)(38,168)(39,169)(40,170)(41,171)(42,172)(43,173)(44,174)(45,175)(46,176)(47,84)(48,85)(49,86)(50,87)(51,88)(52,89)(53,90)(54,91)(55,92)(56,70)(57,71)(58,72)(59,73)(60,74)(61,75)(62,76)(63,77)(64,78)(65,79)(66,80)(67,81)(68,82)(69,83)(93,139)(94,140)(95,141)(96,142)(97,143)(98,144)(99,145)(100,146)(101,147)(102,148)(103,149)(104,150)(105,151)(106,152)(107,153)(108,154)(109,155)(110,156)(111,157)(112,158)(113,159)(114,160)(115,161) );
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)], [(1,184,52,112,158,89,31,132),(2,162,53,113,159,90,32,133),(3,163,54,114,160,91,33,134),(4,164,55,115,161,92,34,135),(5,165,56,93,139,70,35,136),(6,166,57,94,140,71,36,137),(7,167,58,95,141,72,37,138),(8,168,59,96,142,73,38,116),(9,169,60,97,143,74,39,117),(10,170,61,98,144,75,40,118),(11,171,62,99,145,76,41,119),(12,172,63,100,146,77,42,120),(13,173,64,101,147,78,43,121),(14,174,65,102,148,79,44,122),(15,175,66,103,149,80,45,123),(16,176,67,104,150,81,46,124),(17,177,68,105,151,82,24,125),(18,178,69,106,152,83,25,126),(19,179,47,107,153,84,26,127),(20,180,48,108,154,85,27,128),(21,181,49,109,155,86,28,129),(22,182,50,110,156,87,29,130),(23,183,51,111,157,88,30,131)], [(1,132),(2,133),(3,134),(4,135),(5,136),(6,137),(7,138),(8,116),(9,117),(10,118),(11,119),(12,120),(13,121),(14,122),(15,123),(16,124),(17,125),(18,126),(19,127),(20,128),(21,129),(22,130),(23,131),(24,177),(25,178),(26,179),(27,180),(28,181),(29,182),(30,183),(31,184),(32,162),(33,163),(34,164),(35,165),(36,166),(37,167),(38,168),(39,169),(40,170),(41,171),(42,172),(43,173),(44,174),(45,175),(46,176),(47,84),(48,85),(49,86),(50,87),(51,88),(52,89),(53,90),(54,91),(55,92),(56,70),(57,71),(58,72),(59,73),(60,74),(61,75),(62,76),(63,77),(64,78),(65,79),(66,80),(67,81),(68,82),(69,83),(93,139),(94,140),(95,141),(96,142),(97,143),(98,144),(99,145),(100,146),(101,147),(102,148),(103,149),(104,150),(105,151),(106,152),(107,153),(108,154),(109,155),(110,156),(111,157),(112,158),(113,159),(114,160),(115,161)]])
161 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4 | 8A | 8B | 23A | ··· | 23V | 46A | ··· | 46V | 46W | ··· | 46BN | 92A | ··· | 92V | 184A | ··· | 184AR |
| order | 1 | 2 | 2 | 2 | 4 | 8 | 8 | 23 | ··· | 23 | 46 | ··· | 46 | 46 | ··· | 46 | 92 | ··· | 92 | 184 | ··· | 184 |
| size | 1 | 1 | 4 | 4 | 2 | 2 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 4 | ··· | 4 | 2 | ··· | 2 | 2 | ··· | 2 |
161 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C23 | C46 | C46 | D4 | D8 | D4×C23 | D8×C23 |
| kernel | D8×C23 | C184 | D4×C23 | D8 | C8 | D4 | C46 | C23 | C2 | C1 |
| # reps | 1 | 1 | 2 | 22 | 22 | 44 | 1 | 2 | 22 | 44 |
Matrix representation of D8×C23 ►in GL2(𝔽47) generated by
| 17 | 0 |
| 0 | 17 |
| 7 | 6 |
| 39 | 0 |
| 0 | 6 |
| 8 | 0 |
G:=sub<GL(2,GF(47))| [17,0,0,17],[7,39,6,0],[0,8,6,0] >;
D8×C23 in GAP, Magma, Sage, TeX
D_8\times C_{23} % in TeX
G:=Group("D8xC23"); // GroupNames label
G:=SmallGroup(368,24);
// by ID
G=gap.SmallGroup(368,24);
# by ID
G:=PCGroup([5,-2,-2,-23,-2,-2,941,5523,2768,58]);
// Polycyclic
G:=Group<a,b,c|a^23=b^8=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export