metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: D51⋊2C4, D102.C2, C6.3D34, C34.3D6, Dic17⋊2S3, Dic3⋊2D17, C102.3C22, C17⋊2(C4×S3), C51⋊6(C2×C4), C3⋊1(C4×D17), C2.3(S3×D17), (Dic3×C17)⋊2C2, (C3×Dic17)⋊2C2, SmallGroup(408,9)
Series: Derived ►Chief ►Lower central ►Upper central
| C51 — D51⋊2C4 |
Generators and relations for D51⋊2C4
G = < a,b,c | a51=b2=c4=1, bab=a-1, cac-1=a35, cbc-1=a34b >
Smallest permutation representation of D51⋊2C4
►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 83)(2 82)(3 81)(4 80)(5 79)(6 78)(7 77)(8 76)(9 75)(10 74)(11 73)(12 72)(13 71)(14 70)(15 69)(16 68)(17 67)(18 66)(19 65)(20 64)(21 63)(22 62)(23 61)(24 60)(25 59)(26 58)(27 57)(28 56)(29 55)(30 54)(31 53)(32 52)(33 102)(34 101)(35 100)(36 99)(37 98)(38 97)(39 96)(40 95)(41 94)(42 93)(43 92)(44 91)(45 90)(46 89)(47 88)(48 87)(49 86)(50 85)(51 84)(103 176)(104 175)(105 174)(106 173)(107 172)(108 171)(109 170)(110 169)(111 168)(112 167)(113 166)(114 165)(115 164)(116 163)(117 162)(118 161)(119 160)(120 159)(121 158)(122 157)(123 156)(124 155)(125 154)(126 204)(127 203)(128 202)(129 201)(130 200)(131 199)(132 198)(133 197)(134 196)(135 195)(136 194)(137 193)(138 192)(139 191)(140 190)(141 189)(142 188)(143 187)(144 186)(145 185)(146 184)(147 183)(148 182)(149 181)(150 180)(151 179)(152 178)(153 177)
(1 204 84 127)(2 188 85 111)(3 172 86 146)(4 156 87 130)(5 191 88 114)(6 175 89 149)(7 159 90 133)(8 194 91 117)(9 178 92 152)(10 162 93 136)(11 197 94 120)(12 181 95 104)(13 165 96 139)(14 200 97 123)(15 184 98 107)(16 168 99 142)(17 203 100 126)(18 187 101 110)(19 171 102 145)(20 155 52 129)(21 190 53 113)(22 174 54 148)(23 158 55 132)(24 193 56 116)(25 177 57 151)(26 161 58 135)(27 196 59 119)(28 180 60 103)(29 164 61 138)(30 199 62 122)(31 183 63 106)(32 167 64 141)(33 202 65 125)(34 186 66 109)(35 170 67 144)(36 154 68 128)(37 189 69 112)(38 173 70 147)(39 157 71 131)(40 192 72 115)(41 176 73 150)(42 160 74 134)(43 195 75 118)(44 179 76 153)(45 163 77 137)(46 198 78 121)(47 182 79 105)(48 166 80 140)(49 201 81 124)(50 185 82 108)(51 169 83 143)
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), (1,83)(2,82)(3,81)(4,80)(5,79)(6,78)(7,77)(8,76)(9,75)(10,74)(11,73)(12,72)(13,71)(14,70)(15,69)(16,68)(17,67)(18,66)(19,65)(20,64)(21,63)(22,62)(23,61)(24,60)(25,59)(26,58)(27,57)(28,56)(29,55)(30,54)(31,53)(32,52)(33,102)(34,101)(35,100)(36,99)(37,98)(38,97)(39,96)(40,95)(41,94)(42,93)(43,92)(44,91)(45,90)(46,89)(47,88)(48,87)(49,86)(50,85)(51,84)(103,176)(104,175)(105,174)(106,173)(107,172)(108,171)(109,170)(110,169)(111,168)(112,167)(113,166)(114,165)(115,164)(116,163)(117,162)(118,161)(119,160)(120,159)(121,158)(122,157)(123,156)(124,155)(125,154)(126,204)(127,203)(128,202)(129,201)(130,200)(131,199)(132,198)(133,197)(134,196)(135,195)(136,194)(137,193)(138,192)(139,191)(140,190)(141,189)(142,188)(143,187)(144,186)(145,185)(146,184)(147,183)(148,182)(149,181)(150,180)(151,179)(152,178)(153,177), (1,204,84,127)(2,188,85,111)(3,172,86,146)(4,156,87,130)(5,191,88,114)(6,175,89,149)(7,159,90,133)(8,194,91,117)(9,178,92,152)(10,162,93,136)(11,197,94,120)(12,181,95,104)(13,165,96,139)(14,200,97,123)(15,184,98,107)(16,168,99,142)(17,203,100,126)(18,187,101,110)(19,171,102,145)(20,155,52,129)(21,190,53,113)(22,174,54,148)(23,158,55,132)(24,193,56,116)(25,177,57,151)(26,161,58,135)(27,196,59,119)(28,180,60,103)(29,164,61,138)(30,199,62,122)(31,183,63,106)(32,167,64,141)(33,202,65,125)(34,186,66,109)(35,170,67,144)(36,154,68,128)(37,189,69,112)(38,173,70,147)(39,157,71,131)(40,192,72,115)(41,176,73,150)(42,160,74,134)(43,195,75,118)(44,179,76,153)(45,163,77,137)(46,198,78,121)(47,182,79,105)(48,166,80,140)(49,201,81,124)(50,185,82,108)(51,169,83,143)>;
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), (1,83)(2,82)(3,81)(4,80)(5,79)(6,78)(7,77)(8,76)(9,75)(10,74)(11,73)(12,72)(13,71)(14,70)(15,69)(16,68)(17,67)(18,66)(19,65)(20,64)(21,63)(22,62)(23,61)(24,60)(25,59)(26,58)(27,57)(28,56)(29,55)(30,54)(31,53)(32,52)(33,102)(34,101)(35,100)(36,99)(37,98)(38,97)(39,96)(40,95)(41,94)(42,93)(43,92)(44,91)(45,90)(46,89)(47,88)(48,87)(49,86)(50,85)(51,84)(103,176)(104,175)(105,174)(106,173)(107,172)(108,171)(109,170)(110,169)(111,168)(112,167)(113,166)(114,165)(115,164)(116,163)(117,162)(118,161)(119,160)(120,159)(121,158)(122,157)(123,156)(124,155)(125,154)(126,204)(127,203)(128,202)(129,201)(130,200)(131,199)(132,198)(133,197)(134,196)(135,195)(136,194)(137,193)(138,192)(139,191)(140,190)(141,189)(142,188)(143,187)(144,186)(145,185)(146,184)(147,183)(148,182)(149,181)(150,180)(151,179)(152,178)(153,177), (1,204,84,127)(2,188,85,111)(3,172,86,146)(4,156,87,130)(5,191,88,114)(6,175,89,149)(7,159,90,133)(8,194,91,117)(9,178,92,152)(10,162,93,136)(11,197,94,120)(12,181,95,104)(13,165,96,139)(14,200,97,123)(15,184,98,107)(16,168,99,142)(17,203,100,126)(18,187,101,110)(19,171,102,145)(20,155,52,129)(21,190,53,113)(22,174,54,148)(23,158,55,132)(24,193,56,116)(25,177,57,151)(26,161,58,135)(27,196,59,119)(28,180,60,103)(29,164,61,138)(30,199,62,122)(31,183,63,106)(32,167,64,141)(33,202,65,125)(34,186,66,109)(35,170,67,144)(36,154,68,128)(37,189,69,112)(38,173,70,147)(39,157,71,131)(40,192,72,115)(41,176,73,150)(42,160,74,134)(43,195,75,118)(44,179,76,153)(45,163,77,137)(46,198,78,121)(47,182,79,105)(48,166,80,140)(49,201,81,124)(50,185,82,108)(51,169,83,143) );
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)], [(1,83),(2,82),(3,81),(4,80),(5,79),(6,78),(7,77),(8,76),(9,75),(10,74),(11,73),(12,72),(13,71),(14,70),(15,69),(16,68),(17,67),(18,66),(19,65),(20,64),(21,63),(22,62),(23,61),(24,60),(25,59),(26,58),(27,57),(28,56),(29,55),(30,54),(31,53),(32,52),(33,102),(34,101),(35,100),(36,99),(37,98),(38,97),(39,96),(40,95),(41,94),(42,93),(43,92),(44,91),(45,90),(46,89),(47,88),(48,87),(49,86),(50,85),(51,84),(103,176),(104,175),(105,174),(106,173),(107,172),(108,171),(109,170),(110,169),(111,168),(112,167),(113,166),(114,165),(115,164),(116,163),(117,162),(118,161),(119,160),(120,159),(121,158),(122,157),(123,156),(124,155),(125,154),(126,204),(127,203),(128,202),(129,201),(130,200),(131,199),(132,198),(133,197),(134,196),(135,195),(136,194),(137,193),(138,192),(139,191),(140,190),(141,189),(142,188),(143,187),(144,186),(145,185),(146,184),(147,183),(148,182),(149,181),(150,180),(151,179),(152,178),(153,177)], [(1,204,84,127),(2,188,85,111),(3,172,86,146),(4,156,87,130),(5,191,88,114),(6,175,89,149),(7,159,90,133),(8,194,91,117),(9,178,92,152),(10,162,93,136),(11,197,94,120),(12,181,95,104),(13,165,96,139),(14,200,97,123),(15,184,98,107),(16,168,99,142),(17,203,100,126),(18,187,101,110),(19,171,102,145),(20,155,52,129),(21,190,53,113),(22,174,54,148),(23,158,55,132),(24,193,56,116),(25,177,57,151),(26,161,58,135),(27,196,59,119),(28,180,60,103),(29,164,61,138),(30,199,62,122),(31,183,63,106),(32,167,64,141),(33,202,65,125),(34,186,66,109),(35,170,67,144),(36,154,68,128),(37,189,69,112),(38,173,70,147),(39,157,71,131),(40,192,72,115),(41,176,73,150),(42,160,74,134),(43,195,75,118),(44,179,76,153),(45,163,77,137),(46,198,78,121),(47,182,79,105),(48,166,80,140),(49,201,81,124),(50,185,82,108),(51,169,83,143)]])
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 6 | 12A | 12B | 17A | ··· | 17H | 34A | ··· | 34H | 51A | ··· | 51H | 68A | ··· | 68P | 102A | ··· | 102H |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 6 | 12 | 12 | 17 | ··· | 17 | 34 | ··· | 34 | 51 | ··· | 51 | 68 | ··· | 68 | 102 | ··· | 102 |
| size | 1 | 1 | 51 | 51 | 2 | 3 | 3 | 17 | 17 | 2 | 34 | 34 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 6 | ··· | 6 | 4 | ··· | 4 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C4 | S3 | D6 | C4×S3 | D17 | D34 | C4×D17 | S3×D17 | D51⋊2C4 |
| kernel | D51⋊2C4 | Dic3×C17 | C3×Dic17 | D102 | D51 | Dic17 | C34 | C17 | Dic3 | C6 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 2 | 8 | 8 | 16 | 8 | 8 |
Matrix representation of D51⋊2C4 ►in GL4(𝔽409) generated by
| 272 | 135 | 0 | 0 |
| 274 | 342 | 0 | 0 |
| 0 | 0 | 1 | 50 |
| 0 | 0 | 139 | 407 |
| 137 | 274 | 0 | 0 |
| 239 | 272 | 0 | 0 |
| 0 | 0 | 2 | 50 |
| 0 | 0 | 139 | 407 |
| 143 | 0 | 0 | 0 |
| 0 | 143 | 0 | 0 |
| 0 | 0 | 1 | 50 |
| 0 | 0 | 0 | 408 |
G:=sub<GL(4,GF(409))| [272,274,0,0,135,342,0,0,0,0,1,139,0,0,50,407],[137,239,0,0,274,272,0,0,0,0,2,139,0,0,50,407],[143,0,0,0,0,143,0,0,0,0,1,0,0,0,50,408] >;
D51⋊2C4 in GAP, Magma, Sage, TeX
D_{51}\rtimes_2C_4 % in TeX
G:=Group("D51:2C4"); // GroupNames label
G:=SmallGroup(408,9);
// by ID
G=gap.SmallGroup(408,9);
# by ID
G:=PCGroup([5,-2,-2,-2,-3,-17,20,26,168,9604]);
// Polycyclic
G:=Group<a,b,c|a^51=b^2=c^4=1,b*a*b=a^-1,c*a*c^-1=a^35,c*b*c^-1=a^34*b>;
// generators/relations
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