metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D52⋊6C4, C52.1D4, C4.9D52, C26.7D8, C26.7SD16, C4⋊C4⋊1D13, C4.1(C4×D13), C52.24(C2×C4), (C2×D52).6C2, (C2×C4).35D26, (C2×C26).30D4, C13⋊2(D4⋊C4), C2.2(D4⋊D13), C2.2(Q8⋊D13), (C2×C52).10C22, C2.5(D26⋊C4), C26.14(C22⋊C4), C22.14(C13⋊D4), (C13×C4⋊C4)⋊1C2, (C2×C13⋊2C8)⋊1C2, SmallGroup(416,16)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D52⋊6C4
G = < a,b,c | a52=b2=c4=1, bab=a-1, cac-1=a27, cbc-1=a39b >
52C2
52C2
4C4
26C22
26C22
52C22
52C22
4D13
4D13
13D4
13D4
26C8
26D4
26C23
2D26
2D26
4D26
4D26
4C52
13C2×C8
13C2×D4
2D52
13D4⋊C4
Smallest permutation representation of D52⋊6C4
►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 96)(54 95)(55 94)(56 93)(57 92)(58 91)(59 90)(60 89)(61 88)(62 87)(63 86)(64 85)(65 84)(66 83)(67 82)(68 81)(69 80)(70 79)(71 78)(72 77)(73 76)(74 75)(97 104)(98 103)(99 102)(100 101)(105 127)(106 126)(107 125)(108 124)(109 123)(110 122)(111 121)(112 120)(113 119)(114 118)(115 117)(128 156)(129 155)(130 154)(131 153)(132 152)(133 151)(134 150)(135 149)(136 148)(137 147)(138 146)(139 145)(140 144)(141 143)(158 208)(159 207)(160 206)(161 205)(162 204)(163 203)(164 202)(165 201)(166 200)(167 199)(168 198)(169 197)(170 196)(171 195)(172 194)(173 193)(174 192)(175 191)(176 190)(177 189)(178 188)(179 187)(180 186)(181 185)(182 184)
(1 203 75 136)(2 178 76 111)(3 205 77 138)(4 180 78 113)(5 207 79 140)(6 182 80 115)(7 157 81 142)(8 184 82 117)(9 159 83 144)(10 186 84 119)(11 161 85 146)(12 188 86 121)(13 163 87 148)(14 190 88 123)(15 165 89 150)(16 192 90 125)(17 167 91 152)(18 194 92 127)(19 169 93 154)(20 196 94 129)(21 171 95 156)(22 198 96 131)(23 173 97 106)(24 200 98 133)(25 175 99 108)(26 202 100 135)(27 177 101 110)(28 204 102 137)(29 179 103 112)(30 206 104 139)(31 181 53 114)(32 208 54 141)(33 183 55 116)(34 158 56 143)(35 185 57 118)(36 160 58 145)(37 187 59 120)(38 162 60 147)(39 189 61 122)(40 164 62 149)(41 191 63 124)(42 166 64 151)(43 193 65 126)(44 168 66 153)(45 195 67 128)(46 170 68 155)(47 197 69 130)(48 172 70 105)(49 199 71 132)(50 174 72 107)(51 201 73 134)(52 176 74 109)
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)(53,96)(54,95)(55,94)(56,93)(57,92)(58,91)(59,90)(60,89)(61,88)(62,87)(63,86)(64,85)(65,84)(66,83)(67,82)(68,81)(69,80)(70,79)(71,78)(72,77)(73,76)(74,75)(97,104)(98,103)(99,102)(100,101)(105,127)(106,126)(107,125)(108,124)(109,123)(110,122)(111,121)(112,120)(113,119)(114,118)(115,117)(128,156)(129,155)(130,154)(131,153)(132,152)(133,151)(134,150)(135,149)(136,148)(137,147)(138,146)(139,145)(140,144)(141,143)(158,208)(159,207)(160,206)(161,205)(162,204)(163,203)(164,202)(165,201)(166,200)(167,199)(168,198)(169,197)(170,196)(171,195)(172,194)(173,193)(174,192)(175,191)(176,190)(177,189)(178,188)(179,187)(180,186)(181,185)(182,184), (1,203,75,136)(2,178,76,111)(3,205,77,138)(4,180,78,113)(5,207,79,140)(6,182,80,115)(7,157,81,142)(8,184,82,117)(9,159,83,144)(10,186,84,119)(11,161,85,146)(12,188,86,121)(13,163,87,148)(14,190,88,123)(15,165,89,150)(16,192,90,125)(17,167,91,152)(18,194,92,127)(19,169,93,154)(20,196,94,129)(21,171,95,156)(22,198,96,131)(23,173,97,106)(24,200,98,133)(25,175,99,108)(26,202,100,135)(27,177,101,110)(28,204,102,137)(29,179,103,112)(30,206,104,139)(31,181,53,114)(32,208,54,141)(33,183,55,116)(34,158,56,143)(35,185,57,118)(36,160,58,145)(37,187,59,120)(38,162,60,147)(39,189,61,122)(40,164,62,149)(41,191,63,124)(42,166,64,151)(43,193,65,126)(44,168,66,153)(45,195,67,128)(46,170,68,155)(47,197,69,130)(48,172,70,105)(49,199,71,132)(50,174,72,107)(51,201,73,134)(52,176,74,109)>;
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)(53,96)(54,95)(55,94)(56,93)(57,92)(58,91)(59,90)(60,89)(61,88)(62,87)(63,86)(64,85)(65,84)(66,83)(67,82)(68,81)(69,80)(70,79)(71,78)(72,77)(73,76)(74,75)(97,104)(98,103)(99,102)(100,101)(105,127)(106,126)(107,125)(108,124)(109,123)(110,122)(111,121)(112,120)(113,119)(114,118)(115,117)(128,156)(129,155)(130,154)(131,153)(132,152)(133,151)(134,150)(135,149)(136,148)(137,147)(138,146)(139,145)(140,144)(141,143)(158,208)(159,207)(160,206)(161,205)(162,204)(163,203)(164,202)(165,201)(166,200)(167,199)(168,198)(169,197)(170,196)(171,195)(172,194)(173,193)(174,192)(175,191)(176,190)(177,189)(178,188)(179,187)(180,186)(181,185)(182,184), (1,203,75,136)(2,178,76,111)(3,205,77,138)(4,180,78,113)(5,207,79,140)(6,182,80,115)(7,157,81,142)(8,184,82,117)(9,159,83,144)(10,186,84,119)(11,161,85,146)(12,188,86,121)(13,163,87,148)(14,190,88,123)(15,165,89,150)(16,192,90,125)(17,167,91,152)(18,194,92,127)(19,169,93,154)(20,196,94,129)(21,171,95,156)(22,198,96,131)(23,173,97,106)(24,200,98,133)(25,175,99,108)(26,202,100,135)(27,177,101,110)(28,204,102,137)(29,179,103,112)(30,206,104,139)(31,181,53,114)(32,208,54,141)(33,183,55,116)(34,158,56,143)(35,185,57,118)(36,160,58,145)(37,187,59,120)(38,162,60,147)(39,189,61,122)(40,164,62,149)(41,191,63,124)(42,166,64,151)(43,193,65,126)(44,168,66,153)(45,195,67,128)(46,170,68,155)(47,197,69,130)(48,172,70,105)(49,199,71,132)(50,174,72,107)(51,201,73,134)(52,176,74,109) );
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),(53,96),(54,95),(55,94),(56,93),(57,92),(58,91),(59,90),(60,89),(61,88),(62,87),(63,86),(64,85),(65,84),(66,83),(67,82),(68,81),(69,80),(70,79),(71,78),(72,77),(73,76),(74,75),(97,104),(98,103),(99,102),(100,101),(105,127),(106,126),(107,125),(108,124),(109,123),(110,122),(111,121),(112,120),(113,119),(114,118),(115,117),(128,156),(129,155),(130,154),(131,153),(132,152),(133,151),(134,150),(135,149),(136,148),(137,147),(138,146),(139,145),(140,144),(141,143),(158,208),(159,207),(160,206),(161,205),(162,204),(163,203),(164,202),(165,201),(166,200),(167,199),(168,198),(169,197),(170,196),(171,195),(172,194),(173,193),(174,192),(175,191),(176,190),(177,189),(178,188),(179,187),(180,186),(181,185),(182,184)], [(1,203,75,136),(2,178,76,111),(3,205,77,138),(4,180,78,113),(5,207,79,140),(6,182,80,115),(7,157,81,142),(8,184,82,117),(9,159,83,144),(10,186,84,119),(11,161,85,146),(12,188,86,121),(13,163,87,148),(14,190,88,123),(15,165,89,150),(16,192,90,125),(17,167,91,152),(18,194,92,127),(19,169,93,154),(20,196,94,129),(21,171,95,156),(22,198,96,131),(23,173,97,106),(24,200,98,133),(25,175,99,108),(26,202,100,135),(27,177,101,110),(28,204,102,137),(29,179,103,112),(30,206,104,139),(31,181,53,114),(32,208,54,141),(33,183,55,116),(34,158,56,143),(35,185,57,118),(36,160,58,145),(37,187,59,120),(38,162,60,147),(39,189,61,122),(40,164,62,149),(41,191,63,124),(42,166,64,151),(43,193,65,126),(44,168,66,153),(45,195,67,128),(46,170,68,155),(47,197,69,130),(48,172,70,105),(49,199,71,132),(50,174,72,107),(51,201,73,134),(52,176,74,109)]])
74 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 8A | 8B | 8C | 8D | 13A | ··· | 13F | 26A | ··· | 26R | 52A | ··· | 52AJ |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 13 | ··· | 13 | 26 | ··· | 26 | 52 | ··· | 52 |
| size | 1 | 1 | 1 | 1 | 52 | 52 | 2 | 2 | 4 | 4 | 26 | 26 | 26 | 26 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
74 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C4 | D4 | D4 | D8 | SD16 | D13 | D26 | C4×D13 | D52 | C13⋊D4 | D4⋊D13 | Q8⋊D13 |
| kernel | D52⋊6C4 | C2×C13⋊2C8 | C13×C4⋊C4 | C2×D52 | D52 | C52 | C2×C26 | C26 | C26 | C4⋊C4 | C2×C4 | C4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 2 | 2 | 6 | 6 | 12 | 12 | 12 | 6 | 6 |
Matrix representation of D52⋊6C4 ►in GL4(𝔽313) generated by
| 211 | 288 | 0 | 0 |
| 77 | 25 | 0 | 0 |
| 0 | 0 | 91 | 52 |
| 0 | 0 | 238 | 222 |
| 293 | 312 | 0 | 0 |
| 86 | 20 | 0 | 0 |
| 0 | 0 | 91 | 52 |
| 0 | 0 | 226 | 222 |
| 307 | 42 | 0 | 0 |
| 96 | 6 | 0 | 0 |
| 0 | 0 | 114 | 10 |
| 0 | 0 | 109 | 199 |
G:=sub<GL(4,GF(313))| [211,77,0,0,288,25,0,0,0,0,91,238,0,0,52,222],[293,86,0,0,312,20,0,0,0,0,91,226,0,0,52,222],[307,96,0,0,42,6,0,0,0,0,114,109,0,0,10,199] >;
D52⋊6C4 in GAP, Magma, Sage, TeX
D_{52}\rtimes_6C_4 % in TeX
G:=Group("D52:6C4"); // GroupNames label
G:=SmallGroup(416,16);
// by ID
G=gap.SmallGroup(416,16);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-13,121,31,579,297,69,13829]);
// 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^39*b>;
// generators/relations
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