metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D52⋊8C4, Dic13⋊5D4, C13⋊4(C4×D4), C52⋊5(C2×C4), C4⋊C4⋊8D13, C4⋊1(C4×D13), D26⋊4(C2×C4), C2.4(D4×D13), (C2×D52).8C2, C26.24(C2×D4), (C2×C4).31D26, (C4×Dic13)⋊3C2, D26⋊C4⋊12C2, C26.33(C4○D4), C26.24(C22×C4), (C2×C26).34C23, (C2×C52).24C22, C2.2(D52⋊C2), C22.18(C22×D13), (C2×Dic13).63C22, (C22×D13).23C22, (C13×C4⋊C4)⋊4C2, (C2×C4×D13)⋊12C2, C2.13(C2×C4×D13), SmallGroup(416,114)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D52⋊8C4
G = < a,b,c | a52=b2=c4=1, bab=a-1, cac-1=a27, cbc-1=a26b >
Subgroups: 688 in 94 conjugacy classes, 43 normal (19 characteristic)
C1, C2, C2, C4, C4, C22, C22, C2×C4, C2×C4, C2×C4, D4, C23, C13, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, D13, C26, C4×D4, Dic13, Dic13, C52, C52, D26, D26, C2×C26, C4×D13, D52, C2×Dic13, C2×C52, C2×C52, C22×D13, C4×Dic13, D26⋊C4, C13×C4⋊C4, C2×C4×D13, C2×D52, D52⋊8C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22×C4, C2×D4, C4○D4, D13, C4×D4, D26, C4×D13, C22×D13, C2×C4×D13, D4×D13, D52⋊C2, D52⋊8C4
►On 208 points
Generators in S208
(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)(157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208)
(1 52)(2 51)(3 50)(4 49)(5 48)(6 47)(7 46)(8 45)(9 44)(10 43)(11 42)(12 41)(13 40)(14 39)(15 38)(16 37)(17 36)(18 35)(19 34)(20 33)(21 32)(22 31)(23 30)(24 29)(25 28)(26 27)(53 84)(54 83)(55 82)(56 81)(57 80)(58 79)(59 78)(60 77)(61 76)(62 75)(63 74)(64 73)(65 72)(66 71)(67 70)(68 69)(85 104)(86 103)(87 102)(88 101)(89 100)(90 99)(91 98)(92 97)(93 96)(94 95)(105 110)(106 109)(107 108)(111 156)(112 155)(113 154)(114 153)(115 152)(116 151)(117 150)(118 149)(119 148)(120 147)(121 146)(122 145)(123 144)(124 143)(125 142)(126 141)(127 140)(128 139)(129 138)(130 137)(131 136)(132 135)(133 134)(157 158)(159 208)(160 207)(161 206)(162 205)(163 204)(164 203)(165 202)(166 201)(167 200)(168 199)(169 198)(170 197)(171 196)(172 195)(173 194)(174 193)(175 192)(176 191)(177 190)(178 189)(179 188)(180 187)(181 186)(182 185)(183 184)
(1 95 184 134)(2 70 185 109)(3 97 186 136)(4 72 187 111)(5 99 188 138)(6 74 189 113)(7 101 190 140)(8 76 191 115)(9 103 192 142)(10 78 193 117)(11 53 194 144)(12 80 195 119)(13 55 196 146)(14 82 197 121)(15 57 198 148)(16 84 199 123)(17 59 200 150)(18 86 201 125)(19 61 202 152)(20 88 203 127)(21 63 204 154)(22 90 205 129)(23 65 206 156)(24 92 207 131)(25 67 208 106)(26 94 157 133)(27 69 158 108)(28 96 159 135)(29 71 160 110)(30 98 161 137)(31 73 162 112)(32 100 163 139)(33 75 164 114)(34 102 165 141)(35 77 166 116)(36 104 167 143)(37 79 168 118)(38 54 169 145)(39 81 170 120)(40 56 171 147)(41 83 172 122)(42 58 173 149)(43 85 174 124)(44 60 175 151)(45 87 176 126)(46 62 177 153)(47 89 178 128)(48 64 179 155)(49 91 180 130)(50 66 181 105)(51 93 182 132)(52 68 183 107)
G:=sub<Sym(208)| (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)(157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208), (1,52)(2,51)(3,50)(4,49)(5,48)(6,47)(7,46)(8,45)(9,44)(10,43)(11,42)(12,41)(13,40)(14,39)(15,38)(16,37)(17,36)(18,35)(19,34)(20,33)(21,32)(22,31)(23,30)(24,29)(25,28)(26,27)(53,84)(54,83)(55,82)(56,81)(57,80)(58,79)(59,78)(60,77)(61,76)(62,75)(63,74)(64,73)(65,72)(66,71)(67,70)(68,69)(85,104)(86,103)(87,102)(88,101)(89,100)(90,99)(91,98)(92,97)(93,96)(94,95)(105,110)(106,109)(107,108)(111,156)(112,155)(113,154)(114,153)(115,152)(116,151)(117,150)(118,149)(119,148)(120,147)(121,146)(122,145)(123,144)(124,143)(125,142)(126,141)(127,140)(128,139)(129,138)(130,137)(131,136)(132,135)(133,134)(157,158)(159,208)(160,207)(161,206)(162,205)(163,204)(164,203)(165,202)(166,201)(167,200)(168,199)(169,198)(170,197)(171,196)(172,195)(173,194)(174,193)(175,192)(176,191)(177,190)(178,189)(179,188)(180,187)(181,186)(182,185)(183,184), (1,95,184,134)(2,70,185,109)(3,97,186,136)(4,72,187,111)(5,99,188,138)(6,74,189,113)(7,101,190,140)(8,76,191,115)(9,103,192,142)(10,78,193,117)(11,53,194,144)(12,80,195,119)(13,55,196,146)(14,82,197,121)(15,57,198,148)(16,84,199,123)(17,59,200,150)(18,86,201,125)(19,61,202,152)(20,88,203,127)(21,63,204,154)(22,90,205,129)(23,65,206,156)(24,92,207,131)(25,67,208,106)(26,94,157,133)(27,69,158,108)(28,96,159,135)(29,71,160,110)(30,98,161,137)(31,73,162,112)(32,100,163,139)(33,75,164,114)(34,102,165,141)(35,77,166,116)(36,104,167,143)(37,79,168,118)(38,54,169,145)(39,81,170,120)(40,56,171,147)(41,83,172,122)(42,58,173,149)(43,85,174,124)(44,60,175,151)(45,87,176,126)(46,62,177,153)(47,89,178,128)(48,64,179,155)(49,91,180,130)(50,66,181,105)(51,93,182,132)(52,68,183,107)>;
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)(157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208), (1,52)(2,51)(3,50)(4,49)(5,48)(6,47)(7,46)(8,45)(9,44)(10,43)(11,42)(12,41)(13,40)(14,39)(15,38)(16,37)(17,36)(18,35)(19,34)(20,33)(21,32)(22,31)(23,30)(24,29)(25,28)(26,27)(53,84)(54,83)(55,82)(56,81)(57,80)(58,79)(59,78)(60,77)(61,76)(62,75)(63,74)(64,73)(65,72)(66,71)(67,70)(68,69)(85,104)(86,103)(87,102)(88,101)(89,100)(90,99)(91,98)(92,97)(93,96)(94,95)(105,110)(106,109)(107,108)(111,156)(112,155)(113,154)(114,153)(115,152)(116,151)(117,150)(118,149)(119,148)(120,147)(121,146)(122,145)(123,144)(124,143)(125,142)(126,141)(127,140)(128,139)(129,138)(130,137)(131,136)(132,135)(133,134)(157,158)(159,208)(160,207)(161,206)(162,205)(163,204)(164,203)(165,202)(166,201)(167,200)(168,199)(169,198)(170,197)(171,196)(172,195)(173,194)(174,193)(175,192)(176,191)(177,190)(178,189)(179,188)(180,187)(181,186)(182,185)(183,184), (1,95,184,134)(2,70,185,109)(3,97,186,136)(4,72,187,111)(5,99,188,138)(6,74,189,113)(7,101,190,140)(8,76,191,115)(9,103,192,142)(10,78,193,117)(11,53,194,144)(12,80,195,119)(13,55,196,146)(14,82,197,121)(15,57,198,148)(16,84,199,123)(17,59,200,150)(18,86,201,125)(19,61,202,152)(20,88,203,127)(21,63,204,154)(22,90,205,129)(23,65,206,156)(24,92,207,131)(25,67,208,106)(26,94,157,133)(27,69,158,108)(28,96,159,135)(29,71,160,110)(30,98,161,137)(31,73,162,112)(32,100,163,139)(33,75,164,114)(34,102,165,141)(35,77,166,116)(36,104,167,143)(37,79,168,118)(38,54,169,145)(39,81,170,120)(40,56,171,147)(41,83,172,122)(42,58,173,149)(43,85,174,124)(44,60,175,151)(45,87,176,126)(46,62,177,153)(47,89,178,128)(48,64,179,155)(49,91,180,130)(50,66,181,105)(51,93,182,132)(52,68,183,107) );
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),(157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208)], [(1,52),(2,51),(3,50),(4,49),(5,48),(6,47),(7,46),(8,45),(9,44),(10,43),(11,42),(12,41),(13,40),(14,39),(15,38),(16,37),(17,36),(18,35),(19,34),(20,33),(21,32),(22,31),(23,30),(24,29),(25,28),(26,27),(53,84),(54,83),(55,82),(56,81),(57,80),(58,79),(59,78),(60,77),(61,76),(62,75),(63,74),(64,73),(65,72),(66,71),(67,70),(68,69),(85,104),(86,103),(87,102),(88,101),(89,100),(90,99),(91,98),(92,97),(93,96),(94,95),(105,110),(106,109),(107,108),(111,156),(112,155),(113,154),(114,153),(115,152),(116,151),(117,150),(118,149),(119,148),(120,147),(121,146),(122,145),(123,144),(124,143),(125,142),(126,141),(127,140),(128,139),(129,138),(130,137),(131,136),(132,135),(133,134),(157,158),(159,208),(160,207),(161,206),(162,205),(163,204),(164,203),(165,202),(166,201),(167,200),(168,199),(169,198),(170,197),(171,196),(172,195),(173,194),(174,193),(175,192),(176,191),(177,190),(178,189),(179,188),(180,187),(181,186),(182,185),(183,184)], [(1,95,184,134),(2,70,185,109),(3,97,186,136),(4,72,187,111),(5,99,188,138),(6,74,189,113),(7,101,190,140),(8,76,191,115),(9,103,192,142),(10,78,193,117),(11,53,194,144),(12,80,195,119),(13,55,196,146),(14,82,197,121),(15,57,198,148),(16,84,199,123),(17,59,200,150),(18,86,201,125),(19,61,202,152),(20,88,203,127),(21,63,204,154),(22,90,205,129),(23,65,206,156),(24,92,207,131),(25,67,208,106),(26,94,157,133),(27,69,158,108),(28,96,159,135),(29,71,160,110),(30,98,161,137),(31,73,162,112),(32,100,163,139),(33,75,164,114),(34,102,165,141),(35,77,166,116),(36,104,167,143),(37,79,168,118),(38,54,169,145),(39,81,170,120),(40,56,171,147),(41,83,172,122),(42,58,173,149),(43,85,174,124),(44,60,175,151),(45,87,176,126),(46,62,177,153),(47,89,178,128),(48,64,179,155),(49,91,180,130),(50,66,181,105),(51,93,182,132),(52,68,183,107)]])
80 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 13A | ··· | 13F | 26A | ··· | 26R | 52A | ··· | 52AJ |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 13 | ··· | 13 | 26 | ··· | 26 | 52 | ··· | 52 |
| size | 1 | 1 | 1 | 1 | 26 | 26 | 26 | 26 | 2 | ··· | 2 | 13 | 13 | 13 | 13 | 26 | 26 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
80 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | C4○D4 | D13 | D26 | C4×D13 | D4×D13 | D52⋊C2 |
| kernel | D52⋊8C4 | C4×Dic13 | D26⋊C4 | C13×C4⋊C4 | C2×C4×D13 | C2×D52 | D52 | Dic13 | C26 | C4⋊C4 | C2×C4 | C4 | C2 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 8 | 2 | 2 | 6 | 18 | 24 | 6 | 6 |
Matrix representation of D52⋊8C4 ►in GL4(𝔽53) generated by
| 49 | 10 | 0 | 0 |
| 14 | 31 | 0 | 0 |
| 0 | 0 | 18 | 19 |
| 0 | 0 | 8 | 35 |
| 9 | 27 | 0 | 0 |
| 52 | 44 | 0 | 0 |
| 0 | 0 | 18 | 19 |
| 0 | 0 | 36 | 35 |
| 30 | 0 | 0 | 0 |
| 0 | 30 | 0 | 0 |
| 0 | 0 | 30 | 0 |
| 0 | 0 | 38 | 23 |
G:=sub<GL(4,GF(53))| [49,14,0,0,10,31,0,0,0,0,18,8,0,0,19,35],[9,52,0,0,27,44,0,0,0,0,18,36,0,0,19,35],[30,0,0,0,0,30,0,0,0,0,30,38,0,0,0,23] >;
D52⋊8C4 in GAP, Magma, Sage, TeX
D_{52}\rtimes_8C_4 % in TeX
G:=Group("D52:8C4"); // GroupNames label
G:=SmallGroup(416,114);
// by ID
G=gap.SmallGroup(416,114);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-13,217,103,188,50,13829]);
// Polycyclic
G:=Group<a,b,c|a^52=b^2=c^4=1,b*a*b=a^-1,c*a*c^-1=a^27,c*b*c^-1=a^26*b>;
// generators/relations