direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C23⋊D4, C46⋊2D4, C23⋊D23, C22⋊2D46, D46⋊3C22, C46.10C23, Dic23⋊2C22, C23⋊3(C2×D4), (C2×C46)⋊3C22, (C22×C46)⋊2C2, (C2×Dic23)⋊4C2, (C22×D23)⋊3C2, C2.10(C22×D23), SmallGroup(368,36)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C23⋊D4
G = < a,b,c,d | a2=b23=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >
Subgroups: 488 in 54 conjugacy classes, 27 normal (11 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C2×C4, D4, C23, C23, C2×D4, C23, D23, C46, C46, C46, Dic23, D46, D46, C2×C46, C2×C46, C2×C46, C2×Dic23, C23⋊D4, C22×D23, C22×C46, C2×C23⋊D4
Quotients: C1, C2, C22, D4, C23, C2×D4, D23, D46, C23⋊D4, C22×D23, C2×C23⋊D4
►On 184 points
Generators in S184
(1 116)(2 117)(3 118)(4 119)(5 120)(6 121)(7 122)(8 123)(9 124)(10 125)(11 126)(12 127)(13 128)(14 129)(15 130)(16 131)(17 132)(18 133)(19 134)(20 135)(21 136)(22 137)(23 138)(24 93)(25 94)(26 95)(27 96)(28 97)(29 98)(30 99)(31 100)(32 101)(33 102)(34 103)(35 104)(36 105)(37 106)(38 107)(39 108)(40 109)(41 110)(42 111)(43 112)(44 113)(45 114)(46 115)(47 162)(48 163)(49 164)(50 165)(51 166)(52 167)(53 168)(54 169)(55 170)(56 171)(57 172)(58 173)(59 174)(60 175)(61 176)(62 177)(63 178)(64 179)(65 180)(66 181)(67 182)(68 183)(69 184)(70 139)(71 140)(72 141)(73 142)(74 143)(75 144)(76 145)(77 146)(78 147)(79 148)(80 149)(81 150)(82 151)(83 152)(84 153)(85 154)(86 155)(87 156)(88 157)(89 158)(90 159)(91 160)(92 161)
(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 47 24 70)(2 69 25 92)(3 68 26 91)(4 67 27 90)(5 66 28 89)(6 65 29 88)(7 64 30 87)(8 63 31 86)(9 62 32 85)(10 61 33 84)(11 60 34 83)(12 59 35 82)(13 58 36 81)(14 57 37 80)(15 56 38 79)(16 55 39 78)(17 54 40 77)(18 53 41 76)(19 52 42 75)(20 51 43 74)(21 50 44 73)(22 49 45 72)(23 48 46 71)(93 139 116 162)(94 161 117 184)(95 160 118 183)(96 159 119 182)(97 158 120 181)(98 157 121 180)(99 156 122 179)(100 155 123 178)(101 154 124 177)(102 153 125 176)(103 152 126 175)(104 151 127 174)(105 150 128 173)(106 149 129 172)(107 148 130 171)(108 147 131 170)(109 146 132 169)(110 145 133 168)(111 144 134 167)(112 143 135 166)(113 142 136 165)(114 141 137 164)(115 140 138 163)
(2 23)(3 22)(4 21)(5 20)(6 19)(7 18)(8 17)(9 16)(10 15)(11 14)(12 13)(25 46)(26 45)(27 44)(28 43)(29 42)(30 41)(31 40)(32 39)(33 38)(34 37)(35 36)(47 70)(48 92)(49 91)(50 90)(51 89)(52 88)(53 87)(54 86)(55 85)(56 84)(57 83)(58 82)(59 81)(60 80)(61 79)(62 78)(63 77)(64 76)(65 75)(66 74)(67 73)(68 72)(69 71)(94 115)(95 114)(96 113)(97 112)(98 111)(99 110)(100 109)(101 108)(102 107)(103 106)(104 105)(117 138)(118 137)(119 136)(120 135)(121 134)(122 133)(123 132)(124 131)(125 130)(126 129)(127 128)(139 162)(140 184)(141 183)(142 182)(143 181)(144 180)(145 179)(146 178)(147 177)(148 176)(149 175)(150 174)(151 173)(152 172)(153 171)(154 170)(155 169)(156 168)(157 167)(158 166)(159 165)(160 164)(161 163)
G:=sub<Sym(184)| (1,116)(2,117)(3,118)(4,119)(5,120)(6,121)(7,122)(8,123)(9,124)(10,125)(11,126)(12,127)(13,128)(14,129)(15,130)(16,131)(17,132)(18,133)(19,134)(20,135)(21,136)(22,137)(23,138)(24,93)(25,94)(26,95)(27,96)(28,97)(29,98)(30,99)(31,100)(32,101)(33,102)(34,103)(35,104)(36,105)(37,106)(38,107)(39,108)(40,109)(41,110)(42,111)(43,112)(44,113)(45,114)(46,115)(47,162)(48,163)(49,164)(50,165)(51,166)(52,167)(53,168)(54,169)(55,170)(56,171)(57,172)(58,173)(59,174)(60,175)(61,176)(62,177)(63,178)(64,179)(65,180)(66,181)(67,182)(68,183)(69,184)(70,139)(71,140)(72,141)(73,142)(74,143)(75,144)(76,145)(77,146)(78,147)(79,148)(80,149)(81,150)(82,151)(83,152)(84,153)(85,154)(86,155)(87,156)(88,157)(89,158)(90,159)(91,160)(92,161), (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), (1,47,24,70)(2,69,25,92)(3,68,26,91)(4,67,27,90)(5,66,28,89)(6,65,29,88)(7,64,30,87)(8,63,31,86)(9,62,32,85)(10,61,33,84)(11,60,34,83)(12,59,35,82)(13,58,36,81)(14,57,37,80)(15,56,38,79)(16,55,39,78)(17,54,40,77)(18,53,41,76)(19,52,42,75)(20,51,43,74)(21,50,44,73)(22,49,45,72)(23,48,46,71)(93,139,116,162)(94,161,117,184)(95,160,118,183)(96,159,119,182)(97,158,120,181)(98,157,121,180)(99,156,122,179)(100,155,123,178)(101,154,124,177)(102,153,125,176)(103,152,126,175)(104,151,127,174)(105,150,128,173)(106,149,129,172)(107,148,130,171)(108,147,131,170)(109,146,132,169)(110,145,133,168)(111,144,134,167)(112,143,135,166)(113,142,136,165)(114,141,137,164)(115,140,138,163), (2,23)(3,22)(4,21)(5,20)(6,19)(7,18)(8,17)(9,16)(10,15)(11,14)(12,13)(25,46)(26,45)(27,44)(28,43)(29,42)(30,41)(31,40)(32,39)(33,38)(34,37)(35,36)(47,70)(48,92)(49,91)(50,90)(51,89)(52,88)(53,87)(54,86)(55,85)(56,84)(57,83)(58,82)(59,81)(60,80)(61,79)(62,78)(63,77)(64,76)(65,75)(66,74)(67,73)(68,72)(69,71)(94,115)(95,114)(96,113)(97,112)(98,111)(99,110)(100,109)(101,108)(102,107)(103,106)(104,105)(117,138)(118,137)(119,136)(120,135)(121,134)(122,133)(123,132)(124,131)(125,130)(126,129)(127,128)(139,162)(140,184)(141,183)(142,182)(143,181)(144,180)(145,179)(146,178)(147,177)(148,176)(149,175)(150,174)(151,173)(152,172)(153,171)(154,170)(155,169)(156,168)(157,167)(158,166)(159,165)(160,164)(161,163)>;
G:=Group( (1,116)(2,117)(3,118)(4,119)(5,120)(6,121)(7,122)(8,123)(9,124)(10,125)(11,126)(12,127)(13,128)(14,129)(15,130)(16,131)(17,132)(18,133)(19,134)(20,135)(21,136)(22,137)(23,138)(24,93)(25,94)(26,95)(27,96)(28,97)(29,98)(30,99)(31,100)(32,101)(33,102)(34,103)(35,104)(36,105)(37,106)(38,107)(39,108)(40,109)(41,110)(42,111)(43,112)(44,113)(45,114)(46,115)(47,162)(48,163)(49,164)(50,165)(51,166)(52,167)(53,168)(54,169)(55,170)(56,171)(57,172)(58,173)(59,174)(60,175)(61,176)(62,177)(63,178)(64,179)(65,180)(66,181)(67,182)(68,183)(69,184)(70,139)(71,140)(72,141)(73,142)(74,143)(75,144)(76,145)(77,146)(78,147)(79,148)(80,149)(81,150)(82,151)(83,152)(84,153)(85,154)(86,155)(87,156)(88,157)(89,158)(90,159)(91,160)(92,161), (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), (1,47,24,70)(2,69,25,92)(3,68,26,91)(4,67,27,90)(5,66,28,89)(6,65,29,88)(7,64,30,87)(8,63,31,86)(9,62,32,85)(10,61,33,84)(11,60,34,83)(12,59,35,82)(13,58,36,81)(14,57,37,80)(15,56,38,79)(16,55,39,78)(17,54,40,77)(18,53,41,76)(19,52,42,75)(20,51,43,74)(21,50,44,73)(22,49,45,72)(23,48,46,71)(93,139,116,162)(94,161,117,184)(95,160,118,183)(96,159,119,182)(97,158,120,181)(98,157,121,180)(99,156,122,179)(100,155,123,178)(101,154,124,177)(102,153,125,176)(103,152,126,175)(104,151,127,174)(105,150,128,173)(106,149,129,172)(107,148,130,171)(108,147,131,170)(109,146,132,169)(110,145,133,168)(111,144,134,167)(112,143,135,166)(113,142,136,165)(114,141,137,164)(115,140,138,163), (2,23)(3,22)(4,21)(5,20)(6,19)(7,18)(8,17)(9,16)(10,15)(11,14)(12,13)(25,46)(26,45)(27,44)(28,43)(29,42)(30,41)(31,40)(32,39)(33,38)(34,37)(35,36)(47,70)(48,92)(49,91)(50,90)(51,89)(52,88)(53,87)(54,86)(55,85)(56,84)(57,83)(58,82)(59,81)(60,80)(61,79)(62,78)(63,77)(64,76)(65,75)(66,74)(67,73)(68,72)(69,71)(94,115)(95,114)(96,113)(97,112)(98,111)(99,110)(100,109)(101,108)(102,107)(103,106)(104,105)(117,138)(118,137)(119,136)(120,135)(121,134)(122,133)(123,132)(124,131)(125,130)(126,129)(127,128)(139,162)(140,184)(141,183)(142,182)(143,181)(144,180)(145,179)(146,178)(147,177)(148,176)(149,175)(150,174)(151,173)(152,172)(153,171)(154,170)(155,169)(156,168)(157,167)(158,166)(159,165)(160,164)(161,163) );
G=PermutationGroup([[(1,116),(2,117),(3,118),(4,119),(5,120),(6,121),(7,122),(8,123),(9,124),(10,125),(11,126),(12,127),(13,128),(14,129),(15,130),(16,131),(17,132),(18,133),(19,134),(20,135),(21,136),(22,137),(23,138),(24,93),(25,94),(26,95),(27,96),(28,97),(29,98),(30,99),(31,100),(32,101),(33,102),(34,103),(35,104),(36,105),(37,106),(38,107),(39,108),(40,109),(41,110),(42,111),(43,112),(44,113),(45,114),(46,115),(47,162),(48,163),(49,164),(50,165),(51,166),(52,167),(53,168),(54,169),(55,170),(56,171),(57,172),(58,173),(59,174),(60,175),(61,176),(62,177),(63,178),(64,179),(65,180),(66,181),(67,182),(68,183),(69,184),(70,139),(71,140),(72,141),(73,142),(74,143),(75,144),(76,145),(77,146),(78,147),(79,148),(80,149),(81,150),(82,151),(83,152),(84,153),(85,154),(86,155),(87,156),(88,157),(89,158),(90,159),(91,160),(92,161)], [(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)], [(1,47,24,70),(2,69,25,92),(3,68,26,91),(4,67,27,90),(5,66,28,89),(6,65,29,88),(7,64,30,87),(8,63,31,86),(9,62,32,85),(10,61,33,84),(11,60,34,83),(12,59,35,82),(13,58,36,81),(14,57,37,80),(15,56,38,79),(16,55,39,78),(17,54,40,77),(18,53,41,76),(19,52,42,75),(20,51,43,74),(21,50,44,73),(22,49,45,72),(23,48,46,71),(93,139,116,162),(94,161,117,184),(95,160,118,183),(96,159,119,182),(97,158,120,181),(98,157,121,180),(99,156,122,179),(100,155,123,178),(101,154,124,177),(102,153,125,176),(103,152,126,175),(104,151,127,174),(105,150,128,173),(106,149,129,172),(107,148,130,171),(108,147,131,170),(109,146,132,169),(110,145,133,168),(111,144,134,167),(112,143,135,166),(113,142,136,165),(114,141,137,164),(115,140,138,163)], [(2,23),(3,22),(4,21),(5,20),(6,19),(7,18),(8,17),(9,16),(10,15),(11,14),(12,13),(25,46),(26,45),(27,44),(28,43),(29,42),(30,41),(31,40),(32,39),(33,38),(34,37),(35,36),(47,70),(48,92),(49,91),(50,90),(51,89),(52,88),(53,87),(54,86),(55,85),(56,84),(57,83),(58,82),(59,81),(60,80),(61,79),(62,78),(63,77),(64,76),(65,75),(66,74),(67,73),(68,72),(69,71),(94,115),(95,114),(96,113),(97,112),(98,111),(99,110),(100,109),(101,108),(102,107),(103,106),(104,105),(117,138),(118,137),(119,136),(120,135),(121,134),(122,133),(123,132),(124,131),(125,130),(126,129),(127,128),(139,162),(140,184),(141,183),(142,182),(143,181),(144,180),(145,179),(146,178),(147,177),(148,176),(149,175),(150,174),(151,173),(152,172),(153,171),(154,170),(155,169),(156,168),(157,167),(158,166),(159,165),(160,164),(161,163)]])
98 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 23A | ··· | 23K | 46A | ··· | 46BY |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 23 | ··· | 23 | 46 | ··· | 46 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 46 | 46 | 46 | 46 | 2 | ··· | 2 | 2 | ··· | 2 |
98 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | D4 | D23 | D46 | C23⋊D4 |
| kernel | C2×C23⋊D4 | C2×Dic23 | C23⋊D4 | C22×D23 | C22×C46 | C46 | C23 | C22 | C2 |
| # reps | 1 | 1 | 4 | 1 | 1 | 2 | 11 | 33 | 44 |
Matrix representation of C2×C23⋊D4 ►in GL3(𝔽277) generated by
| 276 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 0 | 242 | 17 |
| 0 | 1 | 229 |
| 276 | 0 | 0 |
| 0 | 24 | 3 |
| 0 | 177 | 253 |
| 276 | 0 | 0 |
| 0 | 34 | 266 |
| 0 | 105 | 243 |
G:=sub<GL(3,GF(277))| [276,0,0,0,1,0,0,0,1],[1,0,0,0,242,1,0,17,229],[276,0,0,0,24,177,0,3,253],[276,0,0,0,34,105,0,266,243] >;
C2×C23⋊D4 in GAP, Magma, Sage, TeX
C_2\times C_{23}\rtimes D_4 % in TeX
G:=Group("C2xC23:D4"); // GroupNames label
G:=SmallGroup(368,36);
// by ID
G=gap.SmallGroup(368,36);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-23,182,8804]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^23=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations