direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C2×C4×D23, C92⋊3C22, C46.2C23, C22.9D46, D46.4C22, Dic23⋊3C22, C46⋊1(C2×C4), (C2×C92)⋊5C2, C23⋊1(C22×C4), (C2×Dic23)⋊5C2, (C2×C46).9C22, C2.1(C22×D23), (C22×D23).2C2, SmallGroup(368,28)
Series: Derived ►Chief ►Lower central ►Upper central
| C23 — C2×C4×D23 |
Generators and relations for C2×C4×D23
G = < a,b,c,d | a2=b4=c23=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 472 in 54 conjugacy classes, 35 normal (11 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C2×C4, C2×C4, C23, C22×C4, C23, D23, C46, C46, Dic23, C92, D46, C2×C46, C4×D23, C2×Dic23, C2×C92, C22×D23, C2×C4×D23
Quotients: C1, C2, C4, C22, C2×C4, C23, C22×C4, D23, D46, C4×D23, C22×D23, C2×C4×D23
►On 184 points
Generators in S184
(1 113)(2 114)(3 115)(4 93)(5 94)(6 95)(7 96)(8 97)(9 98)(10 99)(11 100)(12 101)(13 102)(14 103)(15 104)(16 105)(17 106)(18 107)(19 108)(20 109)(21 110)(22 111)(23 112)(24 121)(25 122)(26 123)(27 124)(28 125)(29 126)(30 127)(31 128)(32 129)(33 130)(34 131)(35 132)(36 133)(37 134)(38 135)(39 136)(40 137)(41 138)(42 116)(43 117)(44 118)(45 119)(46 120)(47 150)(48 151)(49 152)(50 153)(51 154)(52 155)(53 156)(54 157)(55 158)(56 159)(57 160)(58 161)(59 139)(60 140)(61 141)(62 142)(63 143)(64 144)(65 145)(66 146)(67 147)(68 148)(69 149)(70 169)(71 170)(72 171)(73 172)(74 173)(75 174)(76 175)(77 176)(78 177)(79 178)(80 179)(81 180)(82 181)(83 182)(84 183)(85 184)(86 162)(87 163)(88 164)(89 165)(90 166)(91 167)(92 168)
(1 71 33 49)(2 72 34 50)(3 73 35 51)(4 74 36 52)(5 75 37 53)(6 76 38 54)(7 77 39 55)(8 78 40 56)(9 79 41 57)(10 80 42 58)(11 81 43 59)(12 82 44 60)(13 83 45 61)(14 84 46 62)(15 85 24 63)(16 86 25 64)(17 87 26 65)(18 88 27 66)(19 89 28 67)(20 90 29 68)(21 91 30 69)(22 92 31 47)(23 70 32 48)(93 173 133 155)(94 174 134 156)(95 175 135 157)(96 176 136 158)(97 177 137 159)(98 178 138 160)(99 179 116 161)(100 180 117 139)(101 181 118 140)(102 182 119 141)(103 183 120 142)(104 184 121 143)(105 162 122 144)(106 163 123 145)(107 164 124 146)(108 165 125 147)(109 166 126 148)(110 167 127 149)(111 168 128 150)(112 169 129 151)(113 170 130 152)(114 171 131 153)(115 172 132 154)
(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 23)(2 22)(3 21)(4 20)(5 19)(6 18)(7 17)(8 16)(9 15)(10 14)(11 13)(24 41)(25 40)(26 39)(27 38)(28 37)(29 36)(30 35)(31 34)(32 33)(42 46)(43 45)(47 50)(48 49)(51 69)(52 68)(53 67)(54 66)(55 65)(56 64)(57 63)(58 62)(59 61)(70 71)(72 92)(73 91)(74 90)(75 89)(76 88)(77 87)(78 86)(79 85)(80 84)(81 83)(93 109)(94 108)(95 107)(96 106)(97 105)(98 104)(99 103)(100 102)(110 115)(111 114)(112 113)(116 120)(117 119)(121 138)(122 137)(123 136)(124 135)(125 134)(126 133)(127 132)(128 131)(129 130)(139 141)(142 161)(143 160)(144 159)(145 158)(146 157)(147 156)(148 155)(149 154)(150 153)(151 152)(162 177)(163 176)(164 175)(165 174)(166 173)(167 172)(168 171)(169 170)(178 184)(179 183)(180 182)
G:=sub<Sym(184)| (1,113)(2,114)(3,115)(4,93)(5,94)(6,95)(7,96)(8,97)(9,98)(10,99)(11,100)(12,101)(13,102)(14,103)(15,104)(16,105)(17,106)(18,107)(19,108)(20,109)(21,110)(22,111)(23,112)(24,121)(25,122)(26,123)(27,124)(28,125)(29,126)(30,127)(31,128)(32,129)(33,130)(34,131)(35,132)(36,133)(37,134)(38,135)(39,136)(40,137)(41,138)(42,116)(43,117)(44,118)(45,119)(46,120)(47,150)(48,151)(49,152)(50,153)(51,154)(52,155)(53,156)(54,157)(55,158)(56,159)(57,160)(58,161)(59,139)(60,140)(61,141)(62,142)(63,143)(64,144)(65,145)(66,146)(67,147)(68,148)(69,149)(70,169)(71,170)(72,171)(73,172)(74,173)(75,174)(76,175)(77,176)(78,177)(79,178)(80,179)(81,180)(82,181)(83,182)(84,183)(85,184)(86,162)(87,163)(88,164)(89,165)(90,166)(91,167)(92,168), (1,71,33,49)(2,72,34,50)(3,73,35,51)(4,74,36,52)(5,75,37,53)(6,76,38,54)(7,77,39,55)(8,78,40,56)(9,79,41,57)(10,80,42,58)(11,81,43,59)(12,82,44,60)(13,83,45,61)(14,84,46,62)(15,85,24,63)(16,86,25,64)(17,87,26,65)(18,88,27,66)(19,89,28,67)(20,90,29,68)(21,91,30,69)(22,92,31,47)(23,70,32,48)(93,173,133,155)(94,174,134,156)(95,175,135,157)(96,176,136,158)(97,177,137,159)(98,178,138,160)(99,179,116,161)(100,180,117,139)(101,181,118,140)(102,182,119,141)(103,183,120,142)(104,184,121,143)(105,162,122,144)(106,163,123,145)(107,164,124,146)(108,165,125,147)(109,166,126,148)(110,167,127,149)(111,168,128,150)(112,169,129,151)(113,170,130,152)(114,171,131,153)(115,172,132,154), (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), (1,23)(2,22)(3,21)(4,20)(5,19)(6,18)(7,17)(8,16)(9,15)(10,14)(11,13)(24,41)(25,40)(26,39)(27,38)(28,37)(29,36)(30,35)(31,34)(32,33)(42,46)(43,45)(47,50)(48,49)(51,69)(52,68)(53,67)(54,66)(55,65)(56,64)(57,63)(58,62)(59,61)(70,71)(72,92)(73,91)(74,90)(75,89)(76,88)(77,87)(78,86)(79,85)(80,84)(81,83)(93,109)(94,108)(95,107)(96,106)(97,105)(98,104)(99,103)(100,102)(110,115)(111,114)(112,113)(116,120)(117,119)(121,138)(122,137)(123,136)(124,135)(125,134)(126,133)(127,132)(128,131)(129,130)(139,141)(142,161)(143,160)(144,159)(145,158)(146,157)(147,156)(148,155)(149,154)(150,153)(151,152)(162,177)(163,176)(164,175)(165,174)(166,173)(167,172)(168,171)(169,170)(178,184)(179,183)(180,182)>;
G:=Group( (1,113)(2,114)(3,115)(4,93)(5,94)(6,95)(7,96)(8,97)(9,98)(10,99)(11,100)(12,101)(13,102)(14,103)(15,104)(16,105)(17,106)(18,107)(19,108)(20,109)(21,110)(22,111)(23,112)(24,121)(25,122)(26,123)(27,124)(28,125)(29,126)(30,127)(31,128)(32,129)(33,130)(34,131)(35,132)(36,133)(37,134)(38,135)(39,136)(40,137)(41,138)(42,116)(43,117)(44,118)(45,119)(46,120)(47,150)(48,151)(49,152)(50,153)(51,154)(52,155)(53,156)(54,157)(55,158)(56,159)(57,160)(58,161)(59,139)(60,140)(61,141)(62,142)(63,143)(64,144)(65,145)(66,146)(67,147)(68,148)(69,149)(70,169)(71,170)(72,171)(73,172)(74,173)(75,174)(76,175)(77,176)(78,177)(79,178)(80,179)(81,180)(82,181)(83,182)(84,183)(85,184)(86,162)(87,163)(88,164)(89,165)(90,166)(91,167)(92,168), (1,71,33,49)(2,72,34,50)(3,73,35,51)(4,74,36,52)(5,75,37,53)(6,76,38,54)(7,77,39,55)(8,78,40,56)(9,79,41,57)(10,80,42,58)(11,81,43,59)(12,82,44,60)(13,83,45,61)(14,84,46,62)(15,85,24,63)(16,86,25,64)(17,87,26,65)(18,88,27,66)(19,89,28,67)(20,90,29,68)(21,91,30,69)(22,92,31,47)(23,70,32,48)(93,173,133,155)(94,174,134,156)(95,175,135,157)(96,176,136,158)(97,177,137,159)(98,178,138,160)(99,179,116,161)(100,180,117,139)(101,181,118,140)(102,182,119,141)(103,183,120,142)(104,184,121,143)(105,162,122,144)(106,163,123,145)(107,164,124,146)(108,165,125,147)(109,166,126,148)(110,167,127,149)(111,168,128,150)(112,169,129,151)(113,170,130,152)(114,171,131,153)(115,172,132,154), (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), (1,23)(2,22)(3,21)(4,20)(5,19)(6,18)(7,17)(8,16)(9,15)(10,14)(11,13)(24,41)(25,40)(26,39)(27,38)(28,37)(29,36)(30,35)(31,34)(32,33)(42,46)(43,45)(47,50)(48,49)(51,69)(52,68)(53,67)(54,66)(55,65)(56,64)(57,63)(58,62)(59,61)(70,71)(72,92)(73,91)(74,90)(75,89)(76,88)(77,87)(78,86)(79,85)(80,84)(81,83)(93,109)(94,108)(95,107)(96,106)(97,105)(98,104)(99,103)(100,102)(110,115)(111,114)(112,113)(116,120)(117,119)(121,138)(122,137)(123,136)(124,135)(125,134)(126,133)(127,132)(128,131)(129,130)(139,141)(142,161)(143,160)(144,159)(145,158)(146,157)(147,156)(148,155)(149,154)(150,153)(151,152)(162,177)(163,176)(164,175)(165,174)(166,173)(167,172)(168,171)(169,170)(178,184)(179,183)(180,182) );
G=PermutationGroup([[(1,113),(2,114),(3,115),(4,93),(5,94),(6,95),(7,96),(8,97),(9,98),(10,99),(11,100),(12,101),(13,102),(14,103),(15,104),(16,105),(17,106),(18,107),(19,108),(20,109),(21,110),(22,111),(23,112),(24,121),(25,122),(26,123),(27,124),(28,125),(29,126),(30,127),(31,128),(32,129),(33,130),(34,131),(35,132),(36,133),(37,134),(38,135),(39,136),(40,137),(41,138),(42,116),(43,117),(44,118),(45,119),(46,120),(47,150),(48,151),(49,152),(50,153),(51,154),(52,155),(53,156),(54,157),(55,158),(56,159),(57,160),(58,161),(59,139),(60,140),(61,141),(62,142),(63,143),(64,144),(65,145),(66,146),(67,147),(68,148),(69,149),(70,169),(71,170),(72,171),(73,172),(74,173),(75,174),(76,175),(77,176),(78,177),(79,178),(80,179),(81,180),(82,181),(83,182),(84,183),(85,184),(86,162),(87,163),(88,164),(89,165),(90,166),(91,167),(92,168)], [(1,71,33,49),(2,72,34,50),(3,73,35,51),(4,74,36,52),(5,75,37,53),(6,76,38,54),(7,77,39,55),(8,78,40,56),(9,79,41,57),(10,80,42,58),(11,81,43,59),(12,82,44,60),(13,83,45,61),(14,84,46,62),(15,85,24,63),(16,86,25,64),(17,87,26,65),(18,88,27,66),(19,89,28,67),(20,90,29,68),(21,91,30,69),(22,92,31,47),(23,70,32,48),(93,173,133,155),(94,174,134,156),(95,175,135,157),(96,176,136,158),(97,177,137,159),(98,178,138,160),(99,179,116,161),(100,180,117,139),(101,181,118,140),(102,182,119,141),(103,183,120,142),(104,184,121,143),(105,162,122,144),(106,163,123,145),(107,164,124,146),(108,165,125,147),(109,166,126,148),(110,167,127,149),(111,168,128,150),(112,169,129,151),(113,170,130,152),(114,171,131,153),(115,172,132,154)], [(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)], [(1,23),(2,22),(3,21),(4,20),(5,19),(6,18),(7,17),(8,16),(9,15),(10,14),(11,13),(24,41),(25,40),(26,39),(27,38),(28,37),(29,36),(30,35),(31,34),(32,33),(42,46),(43,45),(47,50),(48,49),(51,69),(52,68),(53,67),(54,66),(55,65),(56,64),(57,63),(58,62),(59,61),(70,71),(72,92),(73,91),(74,90),(75,89),(76,88),(77,87),(78,86),(79,85),(80,84),(81,83),(93,109),(94,108),(95,107),(96,106),(97,105),(98,104),(99,103),(100,102),(110,115),(111,114),(112,113),(116,120),(117,119),(121,138),(122,137),(123,136),(124,135),(125,134),(126,133),(127,132),(128,131),(129,130),(139,141),(142,161),(143,160),(144,159),(145,158),(146,157),(147,156),(148,155),(149,154),(150,153),(151,152),(162,177),(163,176),(164,175),(165,174),(166,173),(167,172),(168,171),(169,170),(178,184),(179,183),(180,182)]])
104 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 23A | ··· | 23K | 46A | ··· | 46AG | 92A | ··· | 92AR |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 23 | ··· | 23 | 46 | ··· | 46 | 92 | ··· | 92 |
| size | 1 | 1 | 1 | 1 | 23 | 23 | 23 | 23 | 1 | 1 | 1 | 1 | 23 | 23 | 23 | 23 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
104 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C4 | D23 | D46 | D46 | C4×D23 |
| kernel | C2×C4×D23 | C4×D23 | C2×Dic23 | C2×C92 | C22×D23 | D46 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 8 | 11 | 22 | 11 | 44 |
Matrix representation of C2×C4×D23 ►in GL3(𝔽277) generated by
| 276 | 0 | 0 |
| 0 | 276 | 0 |
| 0 | 0 | 276 |
| 276 | 0 | 0 |
| 0 | 217 | 0 |
| 0 | 0 | 217 |
| 1 | 0 | 0 |
| 0 | 148 | 1 |
| 0 | 180 | 194 |
| 276 | 0 | 0 |
| 0 | 121 | 18 |
| 0 | 110 | 156 |
G:=sub<GL(3,GF(277))| [276,0,0,0,276,0,0,0,276],[276,0,0,0,217,0,0,0,217],[1,0,0,0,148,180,0,1,194],[276,0,0,0,121,110,0,18,156] >;
C2×C4×D23 in GAP, Magma, Sage, TeX
C_2\times C_4\times D_{23} % in TeX
G:=Group("C2xC4xD23"); // GroupNames label
G:=SmallGroup(368,28);
// by ID
G=gap.SmallGroup(368,28);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-23,42,8804]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^4=c^23=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations