direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary
Aliases: D4×C51, C4⋊C102, C68⋊3C6, C204⋊7C2, C12⋊3C34, C22⋊2C102, C102.23C22, (C2×C34)⋊3C6, (C2×C6)⋊1C34, (C2×C102)⋊1C2, C34.6(C2×C6), C6.6(C2×C34), C2.1(C2×C102), SmallGroup(408,31)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4×C51
G = < a,b,c | a51=b4=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of D4×C51
►On 204 points
Generators in S204
(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)(154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204)
(1 111 194 80)(2 112 195 81)(3 113 196 82)(4 114 197 83)(5 115 198 84)(6 116 199 85)(7 117 200 86)(8 118 201 87)(9 119 202 88)(10 120 203 89)(11 121 204 90)(12 122 154 91)(13 123 155 92)(14 124 156 93)(15 125 157 94)(16 126 158 95)(17 127 159 96)(18 128 160 97)(19 129 161 98)(20 130 162 99)(21 131 163 100)(22 132 164 101)(23 133 165 102)(24 134 166 52)(25 135 167 53)(26 136 168 54)(27 137 169 55)(28 138 170 56)(29 139 171 57)(30 140 172 58)(31 141 173 59)(32 142 174 60)(33 143 175 61)(34 144 176 62)(35 145 177 63)(36 146 178 64)(37 147 179 65)(38 148 180 66)(39 149 181 67)(40 150 182 68)(41 151 183 69)(42 152 184 70)(43 153 185 71)(44 103 186 72)(45 104 187 73)(46 105 188 74)(47 106 189 75)(48 107 190 76)(49 108 191 77)(50 109 192 78)(51 110 193 79)
(52 134)(53 135)(54 136)(55 137)(56 138)(57 139)(58 140)(59 141)(60 142)(61 143)(62 144)(63 145)(64 146)(65 147)(66 148)(67 149)(68 150)(69 151)(70 152)(71 153)(72 103)(73 104)(74 105)(75 106)(76 107)(77 108)(78 109)(79 110)(80 111)(81 112)(82 113)(83 114)(84 115)(85 116)(86 117)(87 118)(88 119)(89 120)(90 121)(91 122)(92 123)(93 124)(94 125)(95 126)(96 127)(97 128)(98 129)(99 130)(100 131)(101 132)(102 133)
G:=sub<Sym(204)| (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)(154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204), (1,111,194,80)(2,112,195,81)(3,113,196,82)(4,114,197,83)(5,115,198,84)(6,116,199,85)(7,117,200,86)(8,118,201,87)(9,119,202,88)(10,120,203,89)(11,121,204,90)(12,122,154,91)(13,123,155,92)(14,124,156,93)(15,125,157,94)(16,126,158,95)(17,127,159,96)(18,128,160,97)(19,129,161,98)(20,130,162,99)(21,131,163,100)(22,132,164,101)(23,133,165,102)(24,134,166,52)(25,135,167,53)(26,136,168,54)(27,137,169,55)(28,138,170,56)(29,139,171,57)(30,140,172,58)(31,141,173,59)(32,142,174,60)(33,143,175,61)(34,144,176,62)(35,145,177,63)(36,146,178,64)(37,147,179,65)(38,148,180,66)(39,149,181,67)(40,150,182,68)(41,151,183,69)(42,152,184,70)(43,153,185,71)(44,103,186,72)(45,104,187,73)(46,105,188,74)(47,106,189,75)(48,107,190,76)(49,108,191,77)(50,109,192,78)(51,110,193,79), (52,134)(53,135)(54,136)(55,137)(56,138)(57,139)(58,140)(59,141)(60,142)(61,143)(62,144)(63,145)(64,146)(65,147)(66,148)(67,149)(68,150)(69,151)(70,152)(71,153)(72,103)(73,104)(74,105)(75,106)(76,107)(77,108)(78,109)(79,110)(80,111)(81,112)(82,113)(83,114)(84,115)(85,116)(86,117)(87,118)(88,119)(89,120)(90,121)(91,122)(92,123)(93,124)(94,125)(95,126)(96,127)(97,128)(98,129)(99,130)(100,131)(101,132)(102,133)>;
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)(154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204), (1,111,194,80)(2,112,195,81)(3,113,196,82)(4,114,197,83)(5,115,198,84)(6,116,199,85)(7,117,200,86)(8,118,201,87)(9,119,202,88)(10,120,203,89)(11,121,204,90)(12,122,154,91)(13,123,155,92)(14,124,156,93)(15,125,157,94)(16,126,158,95)(17,127,159,96)(18,128,160,97)(19,129,161,98)(20,130,162,99)(21,131,163,100)(22,132,164,101)(23,133,165,102)(24,134,166,52)(25,135,167,53)(26,136,168,54)(27,137,169,55)(28,138,170,56)(29,139,171,57)(30,140,172,58)(31,141,173,59)(32,142,174,60)(33,143,175,61)(34,144,176,62)(35,145,177,63)(36,146,178,64)(37,147,179,65)(38,148,180,66)(39,149,181,67)(40,150,182,68)(41,151,183,69)(42,152,184,70)(43,153,185,71)(44,103,186,72)(45,104,187,73)(46,105,188,74)(47,106,189,75)(48,107,190,76)(49,108,191,77)(50,109,192,78)(51,110,193,79), (52,134)(53,135)(54,136)(55,137)(56,138)(57,139)(58,140)(59,141)(60,142)(61,143)(62,144)(63,145)(64,146)(65,147)(66,148)(67,149)(68,150)(69,151)(70,152)(71,153)(72,103)(73,104)(74,105)(75,106)(76,107)(77,108)(78,109)(79,110)(80,111)(81,112)(82,113)(83,114)(84,115)(85,116)(86,117)(87,118)(88,119)(89,120)(90,121)(91,122)(92,123)(93,124)(94,125)(95,126)(96,127)(97,128)(98,129)(99,130)(100,131)(101,132)(102,133) );
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),(154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204)], [(1,111,194,80),(2,112,195,81),(3,113,196,82),(4,114,197,83),(5,115,198,84),(6,116,199,85),(7,117,200,86),(8,118,201,87),(9,119,202,88),(10,120,203,89),(11,121,204,90),(12,122,154,91),(13,123,155,92),(14,124,156,93),(15,125,157,94),(16,126,158,95),(17,127,159,96),(18,128,160,97),(19,129,161,98),(20,130,162,99),(21,131,163,100),(22,132,164,101),(23,133,165,102),(24,134,166,52),(25,135,167,53),(26,136,168,54),(27,137,169,55),(28,138,170,56),(29,139,171,57),(30,140,172,58),(31,141,173,59),(32,142,174,60),(33,143,175,61),(34,144,176,62),(35,145,177,63),(36,146,178,64),(37,147,179,65),(38,148,180,66),(39,149,181,67),(40,150,182,68),(41,151,183,69),(42,152,184,70),(43,153,185,71),(44,103,186,72),(45,104,187,73),(46,105,188,74),(47,106,189,75),(48,107,190,76),(49,108,191,77),(50,109,192,78),(51,110,193,79)], [(52,134),(53,135),(54,136),(55,137),(56,138),(57,139),(58,140),(59,141),(60,142),(61,143),(62,144),(63,145),(64,146),(65,147),(66,148),(67,149),(68,150),(69,151),(70,152),(71,153),(72,103),(73,104),(74,105),(75,106),(76,107),(77,108),(78,109),(79,110),(80,111),(81,112),(82,113),(83,114),(84,115),(85,116),(86,117),(87,118),(88,119),(89,120),(90,121),(91,122),(92,123),(93,124),(94,125),(95,126),(96,127),(97,128),(98,129),(99,130),(100,131),(101,132),(102,133)]])
255 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4 | 6A | 6B | 6C | 6D | 6E | 6F | 12A | 12B | 17A | ··· | 17P | 34A | ··· | 34P | 34Q | ··· | 34AV | 51A | ··· | 51AF | 68A | ··· | 68P | 102A | ··· | 102AF | 102AG | ··· | 102CR | 204A | ··· | 204AF |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 17 | ··· | 17 | 34 | ··· | 34 | 34 | ··· | 34 | 51 | ··· | 51 | 68 | ··· | 68 | 102 | ··· | 102 | 102 | ··· | 102 | 204 | ··· | 204 |
| size | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
255 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | ||||||||||||
| image | C1 | C2 | C2 | C3 | C6 | C6 | C17 | C34 | C34 | C51 | C102 | C102 | D4 | C3×D4 | D4×C17 | D4×C51 |
| kernel | D4×C51 | C204 | C2×C102 | D4×C17 | C68 | C2×C34 | C3×D4 | C12 | C2×C6 | D4 | C4 | C22 | C51 | C17 | C3 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 4 | 16 | 16 | 32 | 32 | 32 | 64 | 1 | 2 | 16 | 32 |
Matrix representation of D4×C51 ►in GL2(𝔽409) generated by
| 309 | 0 |
| 0 | 309 |
| 54 | 2 |
| 382 | 355 |
| 1 | 0 |
| 355 | 408 |
G:=sub<GL(2,GF(409))| [309,0,0,309],[54,382,2,355],[1,355,0,408] >;
D4×C51 in GAP, Magma, Sage, TeX
D_4\times C_{51} % in TeX
G:=Group("D4xC51"); // GroupNames label
G:=SmallGroup(408,31);
// by ID
G=gap.SmallGroup(408,31);
# by ID
G:=PCGroup([5,-2,-2,-3,-17,-2,2061]);
// Polycyclic
G:=Group<a,b,c|a^51=b^4=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export