metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D92⋊4C2, Q8⋊2D23, C4.7D46, C46.8C23, C92.7C22, D46.3C22, Dic23.5C22, (C4×D23)⋊3C2, C23⋊3(C4○D4), (Q8×C23)⋊3C2, C2.9(C22×D23), SmallGroup(368,34)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D92⋊C2
G = < a,b,c | a92=b2=c2=1, bab=a-1, cac=a45, cbc=a90b >
Smallest permutation representation of D92⋊C2
►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 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)(93 150)(94 149)(95 148)(96 147)(97 146)(98 145)(99 144)(100 143)(101 142)(102 141)(103 140)(104 139)(105 138)(106 137)(107 136)(108 135)(109 134)(110 133)(111 132)(112 131)(113 130)(114 129)(115 128)(116 127)(117 126)(118 125)(119 124)(120 123)(121 122)(151 184)(152 183)(153 182)(154 181)(155 180)(156 179)(157 178)(158 177)(159 176)(160 175)(161 174)(162 173)(163 172)(164 171)(165 170)(166 169)(167 168)
(1 145)(2 98)(3 143)(4 96)(5 141)(6 94)(7 139)(8 184)(9 137)(10 182)(11 135)(12 180)(13 133)(14 178)(15 131)(16 176)(17 129)(18 174)(19 127)(20 172)(21 125)(22 170)(23 123)(24 168)(25 121)(26 166)(27 119)(28 164)(29 117)(30 162)(31 115)(32 160)(33 113)(34 158)(35 111)(36 156)(37 109)(38 154)(39 107)(40 152)(41 105)(42 150)(43 103)(44 148)(45 101)(46 146)(47 99)(48 144)(49 97)(50 142)(51 95)(52 140)(53 93)(54 138)(55 183)(56 136)(57 181)(58 134)(59 179)(60 132)(61 177)(62 130)(63 175)(64 128)(65 173)(66 126)(67 171)(68 124)(69 169)(70 122)(71 167)(72 120)(73 165)(74 118)(75 163)(76 116)(77 161)(78 114)(79 159)(80 112)(81 157)(82 110)(83 155)(84 108)(85 153)(86 106)(87 151)(88 104)(89 149)(90 102)(91 147)(92 100)
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), (1,92)(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)(93,150)(94,149)(95,148)(96,147)(97,146)(98,145)(99,144)(100,143)(101,142)(102,141)(103,140)(104,139)(105,138)(106,137)(107,136)(108,135)(109,134)(110,133)(111,132)(112,131)(113,130)(114,129)(115,128)(116,127)(117,126)(118,125)(119,124)(120,123)(121,122)(151,184)(152,183)(153,182)(154,181)(155,180)(156,179)(157,178)(158,177)(159,176)(160,175)(161,174)(162,173)(163,172)(164,171)(165,170)(166,169)(167,168), (1,145)(2,98)(3,143)(4,96)(5,141)(6,94)(7,139)(8,184)(9,137)(10,182)(11,135)(12,180)(13,133)(14,178)(15,131)(16,176)(17,129)(18,174)(19,127)(20,172)(21,125)(22,170)(23,123)(24,168)(25,121)(26,166)(27,119)(28,164)(29,117)(30,162)(31,115)(32,160)(33,113)(34,158)(35,111)(36,156)(37,109)(38,154)(39,107)(40,152)(41,105)(42,150)(43,103)(44,148)(45,101)(46,146)(47,99)(48,144)(49,97)(50,142)(51,95)(52,140)(53,93)(54,138)(55,183)(56,136)(57,181)(58,134)(59,179)(60,132)(61,177)(62,130)(63,175)(64,128)(65,173)(66,126)(67,171)(68,124)(69,169)(70,122)(71,167)(72,120)(73,165)(74,118)(75,163)(76,116)(77,161)(78,114)(79,159)(80,112)(81,157)(82,110)(83,155)(84,108)(85,153)(86,106)(87,151)(88,104)(89,149)(90,102)(91,147)(92,100)>;
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), (1,92)(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)(93,150)(94,149)(95,148)(96,147)(97,146)(98,145)(99,144)(100,143)(101,142)(102,141)(103,140)(104,139)(105,138)(106,137)(107,136)(108,135)(109,134)(110,133)(111,132)(112,131)(113,130)(114,129)(115,128)(116,127)(117,126)(118,125)(119,124)(120,123)(121,122)(151,184)(152,183)(153,182)(154,181)(155,180)(156,179)(157,178)(158,177)(159,176)(160,175)(161,174)(162,173)(163,172)(164,171)(165,170)(166,169)(167,168), (1,145)(2,98)(3,143)(4,96)(5,141)(6,94)(7,139)(8,184)(9,137)(10,182)(11,135)(12,180)(13,133)(14,178)(15,131)(16,176)(17,129)(18,174)(19,127)(20,172)(21,125)(22,170)(23,123)(24,168)(25,121)(26,166)(27,119)(28,164)(29,117)(30,162)(31,115)(32,160)(33,113)(34,158)(35,111)(36,156)(37,109)(38,154)(39,107)(40,152)(41,105)(42,150)(43,103)(44,148)(45,101)(46,146)(47,99)(48,144)(49,97)(50,142)(51,95)(52,140)(53,93)(54,138)(55,183)(56,136)(57,181)(58,134)(59,179)(60,132)(61,177)(62,130)(63,175)(64,128)(65,173)(66,126)(67,171)(68,124)(69,169)(70,122)(71,167)(72,120)(73,165)(74,118)(75,163)(76,116)(77,161)(78,114)(79,159)(80,112)(81,157)(82,110)(83,155)(84,108)(85,153)(86,106)(87,151)(88,104)(89,149)(90,102)(91,147)(92,100) );
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)], [(1,92),(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),(93,150),(94,149),(95,148),(96,147),(97,146),(98,145),(99,144),(100,143),(101,142),(102,141),(103,140),(104,139),(105,138),(106,137),(107,136),(108,135),(109,134),(110,133),(111,132),(112,131),(113,130),(114,129),(115,128),(116,127),(117,126),(118,125),(119,124),(120,123),(121,122),(151,184),(152,183),(153,182),(154,181),(155,180),(156,179),(157,178),(158,177),(159,176),(160,175),(161,174),(162,173),(163,172),(164,171),(165,170),(166,169),(167,168)], [(1,145),(2,98),(3,143),(4,96),(5,141),(6,94),(7,139),(8,184),(9,137),(10,182),(11,135),(12,180),(13,133),(14,178),(15,131),(16,176),(17,129),(18,174),(19,127),(20,172),(21,125),(22,170),(23,123),(24,168),(25,121),(26,166),(27,119),(28,164),(29,117),(30,162),(31,115),(32,160),(33,113),(34,158),(35,111),(36,156),(37,109),(38,154),(39,107),(40,152),(41,105),(42,150),(43,103),(44,148),(45,101),(46,146),(47,99),(48,144),(49,97),(50,142),(51,95),(52,140),(53,93),(54,138),(55,183),(56,136),(57,181),(58,134),(59,179),(60,132),(61,177),(62,130),(63,175),(64,128),(65,173),(66,126),(67,171),(68,124),(69,169),(70,122),(71,167),(72,120),(73,165),(74,118),(75,163),(76,116),(77,161),(78,114),(79,159),(80,112),(81,157),(82,110),(83,155),(84,108),(85,153),(86,106),(87,151),(88,104),(89,149),(90,102),(91,147),(92,100)]])
65 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 23A | ··· | 23K | 46A | ··· | 46K | 92A | ··· | 92AG |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 23 | ··· | 23 | 46 | ··· | 46 | 92 | ··· | 92 |
| size | 1 | 1 | 46 | 46 | 46 | 2 | 2 | 2 | 23 | 23 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
65 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C4○D4 | D23 | D46 | D92⋊C2 |
| kernel | D92⋊C2 | C4×D23 | D92 | Q8×C23 | C23 | Q8 | C4 | C1 |
| # reps | 1 | 3 | 3 | 1 | 2 | 11 | 33 | 11 |
Matrix representation of D92⋊C2 ►in GL4(𝔽277) generated by
| 94 | 105 | 0 | 0 |
| 21 | 271 | 0 | 0 |
| 0 | 0 | 276 | 42 |
| 0 | 0 | 211 | 1 |
| 180 | 85 | 0 | 0 |
| 49 | 97 | 0 | 0 |
| 0 | 0 | 276 | 42 |
| 0 | 0 | 0 | 1 |
| 182 | 170 | 0 | 0 |
| 162 | 95 | 0 | 0 |
| 0 | 0 | 217 | 27 |
| 0 | 0 | 195 | 60 |
G:=sub<GL(4,GF(277))| [94,21,0,0,105,271,0,0,0,0,276,211,0,0,42,1],[180,49,0,0,85,97,0,0,0,0,276,0,0,0,42,1],[182,162,0,0,170,95,0,0,0,0,217,195,0,0,27,60] >;
D92⋊C2 in GAP, Magma, Sage, TeX
D_{92}\rtimes C_2 % in TeX
G:=Group("D92:C2"); // GroupNames label
G:=SmallGroup(368,34);
// by ID
G=gap.SmallGroup(368,34);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-23,46,182,97,42,8804]);
// Polycyclic
G:=Group<a,b,c|a^92=b^2=c^2=1,b*a*b=a^-1,c*a*c=a^45,c*b*c=a^90*b>;
// generators/relations
Export