direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C22⋊C4×C23, C22⋊C92, C23.C46, C46.12D4, (C2×C46)⋊1C4, (C2×C4)⋊1C46, (C2×C92)⋊2C2, C2.1(C2×C92), C2.1(D4×C23), C46.10(C2×C4), C22.2(C2×C46), (C22×C46).1C2, (C2×C46).13C22, SmallGroup(368,20)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C22⋊C4×C23
G = < a,b,c,d | a23=b2=c2=d4=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, cd=dc >
Smallest permutation representation of C22⋊C4×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 76)(2 77)(3 78)(4 79)(5 80)(6 81)(7 82)(8 83)(9 84)(10 85)(11 86)(12 87)(13 88)(14 89)(15 90)(16 91)(17 92)(18 70)(19 71)(20 72)(21 73)(22 74)(23 75)(24 49)(25 50)(26 51)(27 52)(28 53)(29 54)(30 55)(31 56)(32 57)(33 58)(34 59)(35 60)(36 61)(37 62)(38 63)(39 64)(40 65)(41 66)(42 67)(43 68)(44 69)(45 47)(46 48)(93 163)(94 164)(95 165)(96 166)(97 167)(98 168)(99 169)(100 170)(101 171)(102 172)(103 173)(104 174)(105 175)(106 176)(107 177)(108 178)(109 179)(110 180)(111 181)(112 182)(113 183)(114 184)(115 162)(116 147)(117 148)(118 149)(119 150)(120 151)(121 152)(122 153)(123 154)(124 155)(125 156)(126 157)(127 158)(128 159)(129 160)(130 161)(131 139)(132 140)(133 141)(134 142)(135 143)(136 144)(137 145)(138 146)
(1 150)(2 151)(3 152)(4 153)(5 154)(6 155)(7 156)(8 157)(9 158)(10 159)(11 160)(12 161)(13 139)(14 140)(15 141)(16 142)(17 143)(18 144)(19 145)(20 146)(21 147)(22 148)(23 149)(24 171)(25 172)(26 173)(27 174)(28 175)(29 176)(30 177)(31 178)(32 179)(33 180)(34 181)(35 182)(36 183)(37 184)(38 162)(39 163)(40 164)(41 165)(42 166)(43 167)(44 168)(45 169)(46 170)(47 99)(48 100)(49 101)(50 102)(51 103)(52 104)(53 105)(54 106)(55 107)(56 108)(57 109)(58 110)(59 111)(60 112)(61 113)(62 114)(63 115)(64 93)(65 94)(66 95)(67 96)(68 97)(69 98)(70 136)(71 137)(72 138)(73 116)(74 117)(75 118)(76 119)(77 120)(78 121)(79 122)(80 123)(81 124)(82 125)(83 126)(84 127)(85 128)(86 129)(87 130)(88 131)(89 132)(90 133)(91 134)(92 135)
(1 34 76 111)(2 35 77 112)(3 36 78 113)(4 37 79 114)(5 38 80 115)(6 39 81 93)(7 40 82 94)(8 41 83 95)(9 42 84 96)(10 43 85 97)(11 44 86 98)(12 45 87 99)(13 46 88 100)(14 24 89 101)(15 25 90 102)(16 26 91 103)(17 27 92 104)(18 28 70 105)(19 29 71 106)(20 30 72 107)(21 31 73 108)(22 32 74 109)(23 33 75 110)(47 161 169 130)(48 139 170 131)(49 140 171 132)(50 141 172 133)(51 142 173 134)(52 143 174 135)(53 144 175 136)(54 145 176 137)(55 146 177 138)(56 147 178 116)(57 148 179 117)(58 149 180 118)(59 150 181 119)(60 151 182 120)(61 152 183 121)(62 153 184 122)(63 154 162 123)(64 155 163 124)(65 156 164 125)(66 157 165 126)(67 158 166 127)(68 159 167 128)(69 160 168 129)
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), (1,76)(2,77)(3,78)(4,79)(5,80)(6,81)(7,82)(8,83)(9,84)(10,85)(11,86)(12,87)(13,88)(14,89)(15,90)(16,91)(17,92)(18,70)(19,71)(20,72)(21,73)(22,74)(23,75)(24,49)(25,50)(26,51)(27,52)(28,53)(29,54)(30,55)(31,56)(32,57)(33,58)(34,59)(35,60)(36,61)(37,62)(38,63)(39,64)(40,65)(41,66)(42,67)(43,68)(44,69)(45,47)(46,48)(93,163)(94,164)(95,165)(96,166)(97,167)(98,168)(99,169)(100,170)(101,171)(102,172)(103,173)(104,174)(105,175)(106,176)(107,177)(108,178)(109,179)(110,180)(111,181)(112,182)(113,183)(114,184)(115,162)(116,147)(117,148)(118,149)(119,150)(120,151)(121,152)(122,153)(123,154)(124,155)(125,156)(126,157)(127,158)(128,159)(129,160)(130,161)(131,139)(132,140)(133,141)(134,142)(135,143)(136,144)(137,145)(138,146), (1,150)(2,151)(3,152)(4,153)(5,154)(6,155)(7,156)(8,157)(9,158)(10,159)(11,160)(12,161)(13,139)(14,140)(15,141)(16,142)(17,143)(18,144)(19,145)(20,146)(21,147)(22,148)(23,149)(24,171)(25,172)(26,173)(27,174)(28,175)(29,176)(30,177)(31,178)(32,179)(33,180)(34,181)(35,182)(36,183)(37,184)(38,162)(39,163)(40,164)(41,165)(42,166)(43,167)(44,168)(45,169)(46,170)(47,99)(48,100)(49,101)(50,102)(51,103)(52,104)(53,105)(54,106)(55,107)(56,108)(57,109)(58,110)(59,111)(60,112)(61,113)(62,114)(63,115)(64,93)(65,94)(66,95)(67,96)(68,97)(69,98)(70,136)(71,137)(72,138)(73,116)(74,117)(75,118)(76,119)(77,120)(78,121)(79,122)(80,123)(81,124)(82,125)(83,126)(84,127)(85,128)(86,129)(87,130)(88,131)(89,132)(90,133)(91,134)(92,135), (1,34,76,111)(2,35,77,112)(3,36,78,113)(4,37,79,114)(5,38,80,115)(6,39,81,93)(7,40,82,94)(8,41,83,95)(9,42,84,96)(10,43,85,97)(11,44,86,98)(12,45,87,99)(13,46,88,100)(14,24,89,101)(15,25,90,102)(16,26,91,103)(17,27,92,104)(18,28,70,105)(19,29,71,106)(20,30,72,107)(21,31,73,108)(22,32,74,109)(23,33,75,110)(47,161,169,130)(48,139,170,131)(49,140,171,132)(50,141,172,133)(51,142,173,134)(52,143,174,135)(53,144,175,136)(54,145,176,137)(55,146,177,138)(56,147,178,116)(57,148,179,117)(58,149,180,118)(59,150,181,119)(60,151,182,120)(61,152,183,121)(62,153,184,122)(63,154,162,123)(64,155,163,124)(65,156,164,125)(66,157,165,126)(67,158,166,127)(68,159,167,128)(69,160,168,129)>;
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), (1,76)(2,77)(3,78)(4,79)(5,80)(6,81)(7,82)(8,83)(9,84)(10,85)(11,86)(12,87)(13,88)(14,89)(15,90)(16,91)(17,92)(18,70)(19,71)(20,72)(21,73)(22,74)(23,75)(24,49)(25,50)(26,51)(27,52)(28,53)(29,54)(30,55)(31,56)(32,57)(33,58)(34,59)(35,60)(36,61)(37,62)(38,63)(39,64)(40,65)(41,66)(42,67)(43,68)(44,69)(45,47)(46,48)(93,163)(94,164)(95,165)(96,166)(97,167)(98,168)(99,169)(100,170)(101,171)(102,172)(103,173)(104,174)(105,175)(106,176)(107,177)(108,178)(109,179)(110,180)(111,181)(112,182)(113,183)(114,184)(115,162)(116,147)(117,148)(118,149)(119,150)(120,151)(121,152)(122,153)(123,154)(124,155)(125,156)(126,157)(127,158)(128,159)(129,160)(130,161)(131,139)(132,140)(133,141)(134,142)(135,143)(136,144)(137,145)(138,146), (1,150)(2,151)(3,152)(4,153)(5,154)(6,155)(7,156)(8,157)(9,158)(10,159)(11,160)(12,161)(13,139)(14,140)(15,141)(16,142)(17,143)(18,144)(19,145)(20,146)(21,147)(22,148)(23,149)(24,171)(25,172)(26,173)(27,174)(28,175)(29,176)(30,177)(31,178)(32,179)(33,180)(34,181)(35,182)(36,183)(37,184)(38,162)(39,163)(40,164)(41,165)(42,166)(43,167)(44,168)(45,169)(46,170)(47,99)(48,100)(49,101)(50,102)(51,103)(52,104)(53,105)(54,106)(55,107)(56,108)(57,109)(58,110)(59,111)(60,112)(61,113)(62,114)(63,115)(64,93)(65,94)(66,95)(67,96)(68,97)(69,98)(70,136)(71,137)(72,138)(73,116)(74,117)(75,118)(76,119)(77,120)(78,121)(79,122)(80,123)(81,124)(82,125)(83,126)(84,127)(85,128)(86,129)(87,130)(88,131)(89,132)(90,133)(91,134)(92,135), (1,34,76,111)(2,35,77,112)(3,36,78,113)(4,37,79,114)(5,38,80,115)(6,39,81,93)(7,40,82,94)(8,41,83,95)(9,42,84,96)(10,43,85,97)(11,44,86,98)(12,45,87,99)(13,46,88,100)(14,24,89,101)(15,25,90,102)(16,26,91,103)(17,27,92,104)(18,28,70,105)(19,29,71,106)(20,30,72,107)(21,31,73,108)(22,32,74,109)(23,33,75,110)(47,161,169,130)(48,139,170,131)(49,140,171,132)(50,141,172,133)(51,142,173,134)(52,143,174,135)(53,144,175,136)(54,145,176,137)(55,146,177,138)(56,147,178,116)(57,148,179,117)(58,149,180,118)(59,150,181,119)(60,151,182,120)(61,152,183,121)(62,153,184,122)(63,154,162,123)(64,155,163,124)(65,156,164,125)(66,157,165,126)(67,158,166,127)(68,159,167,128)(69,160,168,129) );
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)], [(1,76),(2,77),(3,78),(4,79),(5,80),(6,81),(7,82),(8,83),(9,84),(10,85),(11,86),(12,87),(13,88),(14,89),(15,90),(16,91),(17,92),(18,70),(19,71),(20,72),(21,73),(22,74),(23,75),(24,49),(25,50),(26,51),(27,52),(28,53),(29,54),(30,55),(31,56),(32,57),(33,58),(34,59),(35,60),(36,61),(37,62),(38,63),(39,64),(40,65),(41,66),(42,67),(43,68),(44,69),(45,47),(46,48),(93,163),(94,164),(95,165),(96,166),(97,167),(98,168),(99,169),(100,170),(101,171),(102,172),(103,173),(104,174),(105,175),(106,176),(107,177),(108,178),(109,179),(110,180),(111,181),(112,182),(113,183),(114,184),(115,162),(116,147),(117,148),(118,149),(119,150),(120,151),(121,152),(122,153),(123,154),(124,155),(125,156),(126,157),(127,158),(128,159),(129,160),(130,161),(131,139),(132,140),(133,141),(134,142),(135,143),(136,144),(137,145),(138,146)], [(1,150),(2,151),(3,152),(4,153),(5,154),(6,155),(7,156),(8,157),(9,158),(10,159),(11,160),(12,161),(13,139),(14,140),(15,141),(16,142),(17,143),(18,144),(19,145),(20,146),(21,147),(22,148),(23,149),(24,171),(25,172),(26,173),(27,174),(28,175),(29,176),(30,177),(31,178),(32,179),(33,180),(34,181),(35,182),(36,183),(37,184),(38,162),(39,163),(40,164),(41,165),(42,166),(43,167),(44,168),(45,169),(46,170),(47,99),(48,100),(49,101),(50,102),(51,103),(52,104),(53,105),(54,106),(55,107),(56,108),(57,109),(58,110),(59,111),(60,112),(61,113),(62,114),(63,115),(64,93),(65,94),(66,95),(67,96),(68,97),(69,98),(70,136),(71,137),(72,138),(73,116),(74,117),(75,118),(76,119),(77,120),(78,121),(79,122),(80,123),(81,124),(82,125),(83,126),(84,127),(85,128),(86,129),(87,130),(88,131),(89,132),(90,133),(91,134),(92,135)], [(1,34,76,111),(2,35,77,112),(3,36,78,113),(4,37,79,114),(5,38,80,115),(6,39,81,93),(7,40,82,94),(8,41,83,95),(9,42,84,96),(10,43,85,97),(11,44,86,98),(12,45,87,99),(13,46,88,100),(14,24,89,101),(15,25,90,102),(16,26,91,103),(17,27,92,104),(18,28,70,105),(19,29,71,106),(20,30,72,107),(21,31,73,108),(22,32,74,109),(23,33,75,110),(47,161,169,130),(48,139,170,131),(49,140,171,132),(50,141,172,133),(51,142,173,134),(52,143,174,135),(53,144,175,136),(54,145,176,137),(55,146,177,138),(56,147,178,116),(57,148,179,117),(58,149,180,118),(59,150,181,119),(60,151,182,120),(61,152,183,121),(62,153,184,122),(63,154,162,123),(64,155,163,124),(65,156,164,125),(66,157,165,126),(67,158,166,127),(68,159,167,128),(69,160,168,129)]])
230 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 23A | ··· | 23V | 46A | ··· | 46BN | 46BO | ··· | 46DF | 92A | ··· | 92CJ |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 23 | ··· | 23 | 46 | ··· | 46 | 46 | ··· | 46 | 92 | ··· | 92 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
230 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | ||||||
| image | C1 | C2 | C2 | C4 | C23 | C46 | C46 | C92 | D4 | D4×C23 |
| kernel | C22⋊C4×C23 | C2×C92 | C22×C46 | C2×C46 | C22⋊C4 | C2×C4 | C23 | C22 | C46 | C2 |
| # reps | 1 | 2 | 1 | 4 | 22 | 44 | 22 | 88 | 2 | 44 |
Matrix representation of C22⋊C4×C23 ►in GL3(𝔽277) generated by
| 1 | 0 | 0 |
| 0 | 236 | 0 |
| 0 | 0 | 236 |
| 1 | 0 | 0 |
| 0 | 1 | 33 |
| 0 | 0 | 276 |
| 1 | 0 | 0 |
| 0 | 276 | 0 |
| 0 | 0 | 276 |
| 60 | 0 | 0 |
| 0 | 33 | 267 |
| 0 | 275 | 244 |
G:=sub<GL(3,GF(277))| [1,0,0,0,236,0,0,0,236],[1,0,0,0,1,0,0,33,276],[1,0,0,0,276,0,0,0,276],[60,0,0,0,33,275,0,267,244] >;
C22⋊C4×C23 in GAP, Magma, Sage, TeX
C_2^2\rtimes C_4\times C_{23} % in TeX
G:=Group("C2^2:C4xC23"); // GroupNames label
G:=SmallGroup(368,20);
// by ID
G=gap.SmallGroup(368,20);
# by ID
G:=PCGroup([5,-2,-2,-23,-2,-2,920,941]);
// Polycyclic
G:=Group<a,b,c,d|a^23=b^2=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,d*b*d^-1=b*c=c*b,c*d=d*c>;
// generators/relations
Export