direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×D92, C4⋊2D46, C46⋊1D4, C92⋊2C22, D46⋊1C22, C46.3C23, C22.10D46, C23⋊1(C2×D4), (C2×C92)⋊3C2, (C2×C4)⋊2D23, (C22×D23)⋊1C2, C2.4(C22×D23), (C2×C46).10C22, SmallGroup(368,29)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×D92
G = < a,b,c | a2=b92=c2=1, ab=ba, ac=ca, cbc=b-1 >
Subgroups: 664 in 54 conjugacy classes, 27 normal (9 characteristic)
C1, C2, C2, C2, C4, C22, C22, C2×C4, D4, C23, C2×D4, C23, D23, C46, C46, C92, D46, D46, C2×C46, D92, C2×C92, C22×D23, C2×D92
Quotients: C1, C2, C22, D4, C23, C2×D4, D23, D46, D92, C22×D23, C2×D92
►On 184 points
Generators in S184
(1 109)(2 110)(3 111)(4 112)(5 113)(6 114)(7 115)(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)(24 132)(25 133)(26 134)(27 135)(28 136)(29 137)(30 138)(31 139)(32 140)(33 141)(34 142)(35 143)(36 144)(37 145)(38 146)(39 147)(40 148)(41 149)(42 150)(43 151)(44 152)(45 153)(46 154)(47 155)(48 156)(49 157)(50 158)(51 159)(52 160)(53 161)(54 162)(55 163)(56 164)(57 165)(58 166)(59 167)(60 168)(61 169)(62 170)(63 171)(64 172)(65 173)(66 174)(67 175)(68 176)(69 177)(70 178)(71 179)(72 180)(73 181)(74 182)(75 183)(76 184)(77 93)(78 94)(79 95)(80 96)(81 97)(82 98)(83 99)(84 100)(85 101)(86 102)(87 103)(88 104)(89 105)(90 106)(91 107)(92 108)
(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 92)(2 91)(3 90)(4 89)(5 88)(6 87)(7 86)(8 85)(9 84)(10 83)(11 82)(12 81)(13 80)(14 79)(15 78)(16 77)(17 76)(18 75)(19 74)(20 73)(21 72)(22 71)(23 70)(24 69)(25 68)(26 67)(27 66)(28 65)(29 64)(30 63)(31 62)(32 61)(33 60)(34 59)(35 58)(36 57)(37 56)(38 55)(39 54)(40 53)(41 52)(42 51)(43 50)(44 49)(45 48)(46 47)(93 124)(94 123)(95 122)(96 121)(97 120)(98 119)(99 118)(100 117)(101 116)(102 115)(103 114)(104 113)(105 112)(106 111)(107 110)(108 109)(125 184)(126 183)(127 182)(128 181)(129 180)(130 179)(131 178)(132 177)(133 176)(134 175)(135 174)(136 173)(137 172)(138 171)(139 170)(140 169)(141 168)(142 167)(143 166)(144 165)(145 164)(146 163)(147 162)(148 161)(149 160)(150 159)(151 158)(152 157)(153 156)(154 155)
G:=sub<Sym(184)| (1,109)(2,110)(3,111)(4,112)(5,113)(6,114)(7,115)(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)(24,132)(25,133)(26,134)(27,135)(28,136)(29,137)(30,138)(31,139)(32,140)(33,141)(34,142)(35,143)(36,144)(37,145)(38,146)(39,147)(40,148)(41,149)(42,150)(43,151)(44,152)(45,153)(46,154)(47,155)(48,156)(49,157)(50,158)(51,159)(52,160)(53,161)(54,162)(55,163)(56,164)(57,165)(58,166)(59,167)(60,168)(61,169)(62,170)(63,171)(64,172)(65,173)(66,174)(67,175)(68,176)(69,177)(70,178)(71,179)(72,180)(73,181)(74,182)(75,183)(76,184)(77,93)(78,94)(79,95)(80,96)(81,97)(82,98)(83,99)(84,100)(85,101)(86,102)(87,103)(88,104)(89,105)(90,106)(91,107)(92,108), (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), (1,92)(2,91)(3,90)(4,89)(5,88)(6,87)(7,86)(8,85)(9,84)(10,83)(11,82)(12,81)(13,80)(14,79)(15,78)(16,77)(17,76)(18,75)(19,74)(20,73)(21,72)(22,71)(23,70)(24,69)(25,68)(26,67)(27,66)(28,65)(29,64)(30,63)(31,62)(32,61)(33,60)(34,59)(35,58)(36,57)(37,56)(38,55)(39,54)(40,53)(41,52)(42,51)(43,50)(44,49)(45,48)(46,47)(93,124)(94,123)(95,122)(96,121)(97,120)(98,119)(99,118)(100,117)(101,116)(102,115)(103,114)(104,113)(105,112)(106,111)(107,110)(108,109)(125,184)(126,183)(127,182)(128,181)(129,180)(130,179)(131,178)(132,177)(133,176)(134,175)(135,174)(136,173)(137,172)(138,171)(139,170)(140,169)(141,168)(142,167)(143,166)(144,165)(145,164)(146,163)(147,162)(148,161)(149,160)(150,159)(151,158)(152,157)(153,156)(154,155)>;
G:=Group( (1,109)(2,110)(3,111)(4,112)(5,113)(6,114)(7,115)(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)(24,132)(25,133)(26,134)(27,135)(28,136)(29,137)(30,138)(31,139)(32,140)(33,141)(34,142)(35,143)(36,144)(37,145)(38,146)(39,147)(40,148)(41,149)(42,150)(43,151)(44,152)(45,153)(46,154)(47,155)(48,156)(49,157)(50,158)(51,159)(52,160)(53,161)(54,162)(55,163)(56,164)(57,165)(58,166)(59,167)(60,168)(61,169)(62,170)(63,171)(64,172)(65,173)(66,174)(67,175)(68,176)(69,177)(70,178)(71,179)(72,180)(73,181)(74,182)(75,183)(76,184)(77,93)(78,94)(79,95)(80,96)(81,97)(82,98)(83,99)(84,100)(85,101)(86,102)(87,103)(88,104)(89,105)(90,106)(91,107)(92,108), (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), (1,92)(2,91)(3,90)(4,89)(5,88)(6,87)(7,86)(8,85)(9,84)(10,83)(11,82)(12,81)(13,80)(14,79)(15,78)(16,77)(17,76)(18,75)(19,74)(20,73)(21,72)(22,71)(23,70)(24,69)(25,68)(26,67)(27,66)(28,65)(29,64)(30,63)(31,62)(32,61)(33,60)(34,59)(35,58)(36,57)(37,56)(38,55)(39,54)(40,53)(41,52)(42,51)(43,50)(44,49)(45,48)(46,47)(93,124)(94,123)(95,122)(96,121)(97,120)(98,119)(99,118)(100,117)(101,116)(102,115)(103,114)(104,113)(105,112)(106,111)(107,110)(108,109)(125,184)(126,183)(127,182)(128,181)(129,180)(130,179)(131,178)(132,177)(133,176)(134,175)(135,174)(136,173)(137,172)(138,171)(139,170)(140,169)(141,168)(142,167)(143,166)(144,165)(145,164)(146,163)(147,162)(148,161)(149,160)(150,159)(151,158)(152,157)(153,156)(154,155) );
G=PermutationGroup([[(1,109),(2,110),(3,111),(4,112),(5,113),(6,114),(7,115),(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),(24,132),(25,133),(26,134),(27,135),(28,136),(29,137),(30,138),(31,139),(32,140),(33,141),(34,142),(35,143),(36,144),(37,145),(38,146),(39,147),(40,148),(41,149),(42,150),(43,151),(44,152),(45,153),(46,154),(47,155),(48,156),(49,157),(50,158),(51,159),(52,160),(53,161),(54,162),(55,163),(56,164),(57,165),(58,166),(59,167),(60,168),(61,169),(62,170),(63,171),(64,172),(65,173),(66,174),(67,175),(68,176),(69,177),(70,178),(71,179),(72,180),(73,181),(74,182),(75,183),(76,184),(77,93),(78,94),(79,95),(80,96),(81,97),(82,98),(83,99),(84,100),(85,101),(86,102),(87,103),(88,104),(89,105),(90,106),(91,107),(92,108)], [(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)], [(1,92),(2,91),(3,90),(4,89),(5,88),(6,87),(7,86),(8,85),(9,84),(10,83),(11,82),(12,81),(13,80),(14,79),(15,78),(16,77),(17,76),(18,75),(19,74),(20,73),(21,72),(22,71),(23,70),(24,69),(25,68),(26,67),(27,66),(28,65),(29,64),(30,63),(31,62),(32,61),(33,60),(34,59),(35,58),(36,57),(37,56),(38,55),(39,54),(40,53),(41,52),(42,51),(43,50),(44,49),(45,48),(46,47),(93,124),(94,123),(95,122),(96,121),(97,120),(98,119),(99,118),(100,117),(101,116),(102,115),(103,114),(104,113),(105,112),(106,111),(107,110),(108,109),(125,184),(126,183),(127,182),(128,181),(129,180),(130,179),(131,178),(132,177),(133,176),(134,175),(135,174),(136,173),(137,172),(138,171),(139,170),(140,169),(141,168),(142,167),(143,166),(144,165),(145,164),(146,163),(147,162),(148,161),(149,160),(150,159),(151,158),(152,157),(153,156),(154,155)]])
98 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 23A | ··· | 23K | 46A | ··· | 46AG | 92A | ··· | 92AR |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 23 | ··· | 23 | 46 | ··· | 46 | 92 | ··· | 92 |
| size | 1 | 1 | 1 | 1 | 46 | 46 | 46 | 46 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
98 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | D4 | D23 | D46 | D46 | D92 |
| kernel | C2×D92 | D92 | C2×C92 | C22×D23 | C46 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 4 | 1 | 2 | 2 | 11 | 22 | 11 | 44 |
Matrix representation of C2×D92 ►in GL3(𝔽277) generated by
| 276 | 0 | 0 |
| 0 | 276 | 0 |
| 0 | 0 | 276 |
| 276 | 0 | 0 |
| 0 | 251 | 98 |
| 0 | 13 | 196 |
| 1 | 0 | 0 |
| 0 | 48 | 222 |
| 0 | 198 | 229 |
G:=sub<GL(3,GF(277))| [276,0,0,0,276,0,0,0,276],[276,0,0,0,251,13,0,98,196],[1,0,0,0,48,198,0,222,229] >;
C2×D92 in GAP, Magma, Sage, TeX
C_2\times D_{92} % in TeX
G:=Group("C2xD92"); // GroupNames label
G:=SmallGroup(368,29);
// by ID
G=gap.SmallGroup(368,29);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-23,182,42,8804]);
// Polycyclic
G:=Group<a,b,c|a^2=b^92=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations