direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: D4×C52, C42⋊4C26, C4⋊C4⋊7C26, C4⋊1(C2×C52), C52⋊9(C2×C4), (C4×C52)⋊11C2, C2.3(D4×C26), C22⋊C4⋊6C26, (C22×C52)⋊4C2, (C22×C4)⋊2C26, C22⋊1(C2×C52), (C2×D4).7C26, C26.66(C2×D4), (D4×C26).14C2, C2.4(C22×C52), C23.7(C2×C26), C26.39(C4○D4), C26.45(C22×C4), (C2×C26).73C23, (C2×C52).121C22, C22.7(C22×C26), (C22×C26).26C22, (C2×C26)⋊7(C2×C4), (C13×C4⋊C4)⋊16C2, C2.2(C13×C4○D4), (C2×C4).15(C2×C26), (C13×C22⋊C4)⋊14C2, SmallGroup(416,179)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4×C52
G = < a,b,c | a52=b4=c2=1, ab=ba, ac=ca, cbc=b-1 >
Subgroups: 124 in 94 conjugacy classes, 64 normal (24 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, C2×C4, D4, C23, C13, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C26, C26, C4×D4, C52, C52, C2×C26, C2×C26, C2×C26, C2×C52, C2×C52, C2×C52, D4×C13, C22×C26, C4×C52, C13×C22⋊C4, C13×C4⋊C4, C22×C52, D4×C26, D4×C52
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C13, C22×C4, C2×D4, C4○D4, C26, C4×D4, C52, C2×C26, C2×C52, D4×C13, C22×C26, C22×C52, D4×C26, C13×C4○D4, D4×C52
►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)
(1 151 192 102)(2 152 193 103)(3 153 194 104)(4 154 195 53)(5 155 196 54)(6 156 197 55)(7 105 198 56)(8 106 199 57)(9 107 200 58)(10 108 201 59)(11 109 202 60)(12 110 203 61)(13 111 204 62)(14 112 205 63)(15 113 206 64)(16 114 207 65)(17 115 208 66)(18 116 157 67)(19 117 158 68)(20 118 159 69)(21 119 160 70)(22 120 161 71)(23 121 162 72)(24 122 163 73)(25 123 164 74)(26 124 165 75)(27 125 166 76)(28 126 167 77)(29 127 168 78)(30 128 169 79)(31 129 170 80)(32 130 171 81)(33 131 172 82)(34 132 173 83)(35 133 174 84)(36 134 175 85)(37 135 176 86)(38 136 177 87)(39 137 178 88)(40 138 179 89)(41 139 180 90)(42 140 181 91)(43 141 182 92)(44 142 183 93)(45 143 184 94)(46 144 185 95)(47 145 186 96)(48 146 187 97)(49 147 188 98)(50 148 189 99)(51 149 190 100)(52 150 191 101)
(53 154)(54 155)(55 156)(56 105)(57 106)(58 107)(59 108)(60 109)(61 110)(62 111)(63 112)(64 113)(65 114)(66 115)(67 116)(68 117)(69 118)(70 119)(71 120)(72 121)(73 122)(74 123)(75 124)(76 125)(77 126)(78 127)(79 128)(80 129)(81 130)(82 131)(83 132)(84 133)(85 134)(86 135)(87 136)(88 137)(89 138)(90 139)(91 140)(92 141)(93 142)(94 143)(95 144)(96 145)(97 146)(98 147)(99 148)(100 149)(101 150)(102 151)(103 152)(104 153)
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), (1,151,192,102)(2,152,193,103)(3,153,194,104)(4,154,195,53)(5,155,196,54)(6,156,197,55)(7,105,198,56)(8,106,199,57)(9,107,200,58)(10,108,201,59)(11,109,202,60)(12,110,203,61)(13,111,204,62)(14,112,205,63)(15,113,206,64)(16,114,207,65)(17,115,208,66)(18,116,157,67)(19,117,158,68)(20,118,159,69)(21,119,160,70)(22,120,161,71)(23,121,162,72)(24,122,163,73)(25,123,164,74)(26,124,165,75)(27,125,166,76)(28,126,167,77)(29,127,168,78)(30,128,169,79)(31,129,170,80)(32,130,171,81)(33,131,172,82)(34,132,173,83)(35,133,174,84)(36,134,175,85)(37,135,176,86)(38,136,177,87)(39,137,178,88)(40,138,179,89)(41,139,180,90)(42,140,181,91)(43,141,182,92)(44,142,183,93)(45,143,184,94)(46,144,185,95)(47,145,186,96)(48,146,187,97)(49,147,188,98)(50,148,189,99)(51,149,190,100)(52,150,191,101), (53,154)(54,155)(55,156)(56,105)(57,106)(58,107)(59,108)(60,109)(61,110)(62,111)(63,112)(64,113)(65,114)(66,115)(67,116)(68,117)(69,118)(70,119)(71,120)(72,121)(73,122)(74,123)(75,124)(76,125)(77,126)(78,127)(79,128)(80,129)(81,130)(82,131)(83,132)(84,133)(85,134)(86,135)(87,136)(88,137)(89,138)(90,139)(91,140)(92,141)(93,142)(94,143)(95,144)(96,145)(97,146)(98,147)(99,148)(100,149)(101,150)(102,151)(103,152)(104,153)>;
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), (1,151,192,102)(2,152,193,103)(3,153,194,104)(4,154,195,53)(5,155,196,54)(6,156,197,55)(7,105,198,56)(8,106,199,57)(9,107,200,58)(10,108,201,59)(11,109,202,60)(12,110,203,61)(13,111,204,62)(14,112,205,63)(15,113,206,64)(16,114,207,65)(17,115,208,66)(18,116,157,67)(19,117,158,68)(20,118,159,69)(21,119,160,70)(22,120,161,71)(23,121,162,72)(24,122,163,73)(25,123,164,74)(26,124,165,75)(27,125,166,76)(28,126,167,77)(29,127,168,78)(30,128,169,79)(31,129,170,80)(32,130,171,81)(33,131,172,82)(34,132,173,83)(35,133,174,84)(36,134,175,85)(37,135,176,86)(38,136,177,87)(39,137,178,88)(40,138,179,89)(41,139,180,90)(42,140,181,91)(43,141,182,92)(44,142,183,93)(45,143,184,94)(46,144,185,95)(47,145,186,96)(48,146,187,97)(49,147,188,98)(50,148,189,99)(51,149,190,100)(52,150,191,101), (53,154)(54,155)(55,156)(56,105)(57,106)(58,107)(59,108)(60,109)(61,110)(62,111)(63,112)(64,113)(65,114)(66,115)(67,116)(68,117)(69,118)(70,119)(71,120)(72,121)(73,122)(74,123)(75,124)(76,125)(77,126)(78,127)(79,128)(80,129)(81,130)(82,131)(83,132)(84,133)(85,134)(86,135)(87,136)(88,137)(89,138)(90,139)(91,140)(92,141)(93,142)(94,143)(95,144)(96,145)(97,146)(98,147)(99,148)(100,149)(101,150)(102,151)(103,152)(104,153) );
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)], [(1,151,192,102),(2,152,193,103),(3,153,194,104),(4,154,195,53),(5,155,196,54),(6,156,197,55),(7,105,198,56),(8,106,199,57),(9,107,200,58),(10,108,201,59),(11,109,202,60),(12,110,203,61),(13,111,204,62),(14,112,205,63),(15,113,206,64),(16,114,207,65),(17,115,208,66),(18,116,157,67),(19,117,158,68),(20,118,159,69),(21,119,160,70),(22,120,161,71),(23,121,162,72),(24,122,163,73),(25,123,164,74),(26,124,165,75),(27,125,166,76),(28,126,167,77),(29,127,168,78),(30,128,169,79),(31,129,170,80),(32,130,171,81),(33,131,172,82),(34,132,173,83),(35,133,174,84),(36,134,175,85),(37,135,176,86),(38,136,177,87),(39,137,178,88),(40,138,179,89),(41,139,180,90),(42,140,181,91),(43,141,182,92),(44,142,183,93),(45,143,184,94),(46,144,185,95),(47,145,186,96),(48,146,187,97),(49,147,188,98),(50,148,189,99),(51,149,190,100),(52,150,191,101)], [(53,154),(54,155),(55,156),(56,105),(57,106),(58,107),(59,108),(60,109),(61,110),(62,111),(63,112),(64,113),(65,114),(66,115),(67,116),(68,117),(69,118),(70,119),(71,120),(72,121),(73,122),(74,123),(75,124),(76,125),(77,126),(78,127),(79,128),(80,129),(81,130),(82,131),(83,132),(84,133),(85,134),(86,135),(87,136),(88,137),(89,138),(90,139),(91,140),(92,141),(93,142),(94,143),(95,144),(96,145),(97,146),(98,147),(99,148),(100,149),(101,150),(102,151),(103,152),(104,153)]])
260 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 13A | ··· | 13L | 26A | ··· | 26AJ | 26AK | ··· | 26CF | 52A | ··· | 52AV | 52AW | ··· | 52EN |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 13 | ··· | 13 | 26 | ··· | 26 | 26 | ··· | 26 | 52 | ··· | 52 | 52 | ··· | 52 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 |
260 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | |||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C13 | C26 | C26 | C26 | C26 | C26 | C52 | D4 | C4○D4 | D4×C13 | C13×C4○D4 |
| kernel | D4×C52 | C4×C52 | C13×C22⋊C4 | C13×C4⋊C4 | C22×C52 | D4×C26 | D4×C13 | C4×D4 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | D4 | C52 | C26 | C4 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 8 | 12 | 12 | 24 | 12 | 24 | 12 | 96 | 2 | 2 | 24 | 24 |
Matrix representation of D4×C52 ►in GL3(𝔽53) generated by
| 30 | 0 | 0 |
| 0 | 27 | 0 |
| 0 | 0 | 27 |
| 1 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 52 | 0 |
| 52 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 52 |
G:=sub<GL(3,GF(53))| [30,0,0,0,27,0,0,0,27],[1,0,0,0,0,52,0,1,0],[52,0,0,0,1,0,0,0,52] >;
D4×C52 in GAP, Magma, Sage, TeX
D_4\times C_{52} % in TeX
G:=Group("D4xC52"); // GroupNames label
G:=SmallGroup(416,179);
// by ID
G=gap.SmallGroup(416,179);
# by ID
G:=PCGroup([6,-2,-2,-2,-13,-2,-2,1248,1273,950]);
// Polycyclic
G:=Group<a,b,c|a^52=b^4=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations