metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C52⋊2D4, D26⋊3D4, C23.8D26, (C2×D4)⋊4D13, (D4×C26)⋊3C2, C13⋊4(C4⋊D4), C4⋊2(C13⋊D4), C52⋊3C4⋊14C2, C26.50(C2×D4), C2.26(D4×D13), (C2×C4).51D26, C26.31(C4○D4), (C2×C52).34C22, (C2×C26).53C23, C23.D13⋊11C2, C2.17(D4⋊2D13), (C22×C26).20C22, C22.60(C22×D13), (C2×Dic13).19C22, (C22×D13).29C22, (C2×C4×D13)⋊2C2, (C2×C13⋊D4)⋊5C2, C2.14(C2×C13⋊D4), SmallGroup(416,159)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C52⋊2D4
G = < a,b,c | a52=b4=c2=1, bab-1=a-1, cac=a25, cbc=b-1 >
Subgroups: 616 in 94 conjugacy classes, 35 normal (19 characteristic)
C1, C2, C2, C4, C4, C22, C22, C2×C4, C2×C4, D4, C23, C23, C13, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×D4, D13, C26, C26, C4⋊D4, Dic13, C52, D26, D26, C2×C26, C2×C26, C4×D13, C2×Dic13, C2×Dic13, C13⋊D4, C2×C52, D4×C13, C22×D13, C22×C26, C52⋊3C4, C23.D13, C2×C4×D13, C2×C13⋊D4, D4×C26, C52⋊2D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, D13, C4⋊D4, D26, C13⋊D4, C22×D13, D4×D13, D4⋊2D13, C2×C13⋊D4, C52⋊2D4
►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)
(1 132 208 85)(2 131 157 84)(3 130 158 83)(4 129 159 82)(5 128 160 81)(6 127 161 80)(7 126 162 79)(8 125 163 78)(9 124 164 77)(10 123 165 76)(11 122 166 75)(12 121 167 74)(13 120 168 73)(14 119 169 72)(15 118 170 71)(16 117 171 70)(17 116 172 69)(18 115 173 68)(19 114 174 67)(20 113 175 66)(21 112 176 65)(22 111 177 64)(23 110 178 63)(24 109 179 62)(25 108 180 61)(26 107 181 60)(27 106 182 59)(28 105 183 58)(29 156 184 57)(30 155 185 56)(31 154 186 55)(32 153 187 54)(33 152 188 53)(34 151 189 104)(35 150 190 103)(36 149 191 102)(37 148 192 101)(38 147 193 100)(39 146 194 99)(40 145 195 98)(41 144 196 97)(42 143 197 96)(43 142 198 95)(44 141 199 94)(45 140 200 93)(46 139 201 92)(47 138 202 91)(48 137 203 90)(49 136 204 89)(50 135 205 88)(51 134 206 87)(52 133 207 86)
(2 26)(3 51)(4 24)(5 49)(6 22)(7 47)(8 20)(9 45)(10 18)(11 43)(12 16)(13 41)(15 39)(17 37)(19 35)(21 33)(23 31)(25 29)(28 52)(30 50)(32 48)(34 46)(36 44)(38 42)(53 112)(54 137)(55 110)(56 135)(57 108)(58 133)(59 106)(60 131)(61 156)(62 129)(63 154)(64 127)(65 152)(66 125)(67 150)(68 123)(69 148)(70 121)(71 146)(72 119)(73 144)(74 117)(75 142)(76 115)(77 140)(78 113)(79 138)(80 111)(81 136)(82 109)(83 134)(84 107)(85 132)(86 105)(87 130)(88 155)(89 128)(90 153)(91 126)(92 151)(93 124)(94 149)(95 122)(96 147)(97 120)(98 145)(99 118)(100 143)(101 116)(102 141)(103 114)(104 139)(157 181)(158 206)(159 179)(160 204)(161 177)(162 202)(163 175)(164 200)(165 173)(166 198)(167 171)(168 196)(170 194)(172 192)(174 190)(176 188)(178 186)(180 184)(183 207)(185 205)(187 203)(189 201)(191 199)(193 197)
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), (1,132,208,85)(2,131,157,84)(3,130,158,83)(4,129,159,82)(5,128,160,81)(6,127,161,80)(7,126,162,79)(8,125,163,78)(9,124,164,77)(10,123,165,76)(11,122,166,75)(12,121,167,74)(13,120,168,73)(14,119,169,72)(15,118,170,71)(16,117,171,70)(17,116,172,69)(18,115,173,68)(19,114,174,67)(20,113,175,66)(21,112,176,65)(22,111,177,64)(23,110,178,63)(24,109,179,62)(25,108,180,61)(26,107,181,60)(27,106,182,59)(28,105,183,58)(29,156,184,57)(30,155,185,56)(31,154,186,55)(32,153,187,54)(33,152,188,53)(34,151,189,104)(35,150,190,103)(36,149,191,102)(37,148,192,101)(38,147,193,100)(39,146,194,99)(40,145,195,98)(41,144,196,97)(42,143,197,96)(43,142,198,95)(44,141,199,94)(45,140,200,93)(46,139,201,92)(47,138,202,91)(48,137,203,90)(49,136,204,89)(50,135,205,88)(51,134,206,87)(52,133,207,86), (2,26)(3,51)(4,24)(5,49)(6,22)(7,47)(8,20)(9,45)(10,18)(11,43)(12,16)(13,41)(15,39)(17,37)(19,35)(21,33)(23,31)(25,29)(28,52)(30,50)(32,48)(34,46)(36,44)(38,42)(53,112)(54,137)(55,110)(56,135)(57,108)(58,133)(59,106)(60,131)(61,156)(62,129)(63,154)(64,127)(65,152)(66,125)(67,150)(68,123)(69,148)(70,121)(71,146)(72,119)(73,144)(74,117)(75,142)(76,115)(77,140)(78,113)(79,138)(80,111)(81,136)(82,109)(83,134)(84,107)(85,132)(86,105)(87,130)(88,155)(89,128)(90,153)(91,126)(92,151)(93,124)(94,149)(95,122)(96,147)(97,120)(98,145)(99,118)(100,143)(101,116)(102,141)(103,114)(104,139)(157,181)(158,206)(159,179)(160,204)(161,177)(162,202)(163,175)(164,200)(165,173)(166,198)(167,171)(168,196)(170,194)(172,192)(174,190)(176,188)(178,186)(180,184)(183,207)(185,205)(187,203)(189,201)(191,199)(193,197)>;
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), (1,132,208,85)(2,131,157,84)(3,130,158,83)(4,129,159,82)(5,128,160,81)(6,127,161,80)(7,126,162,79)(8,125,163,78)(9,124,164,77)(10,123,165,76)(11,122,166,75)(12,121,167,74)(13,120,168,73)(14,119,169,72)(15,118,170,71)(16,117,171,70)(17,116,172,69)(18,115,173,68)(19,114,174,67)(20,113,175,66)(21,112,176,65)(22,111,177,64)(23,110,178,63)(24,109,179,62)(25,108,180,61)(26,107,181,60)(27,106,182,59)(28,105,183,58)(29,156,184,57)(30,155,185,56)(31,154,186,55)(32,153,187,54)(33,152,188,53)(34,151,189,104)(35,150,190,103)(36,149,191,102)(37,148,192,101)(38,147,193,100)(39,146,194,99)(40,145,195,98)(41,144,196,97)(42,143,197,96)(43,142,198,95)(44,141,199,94)(45,140,200,93)(46,139,201,92)(47,138,202,91)(48,137,203,90)(49,136,204,89)(50,135,205,88)(51,134,206,87)(52,133,207,86), (2,26)(3,51)(4,24)(5,49)(6,22)(7,47)(8,20)(9,45)(10,18)(11,43)(12,16)(13,41)(15,39)(17,37)(19,35)(21,33)(23,31)(25,29)(28,52)(30,50)(32,48)(34,46)(36,44)(38,42)(53,112)(54,137)(55,110)(56,135)(57,108)(58,133)(59,106)(60,131)(61,156)(62,129)(63,154)(64,127)(65,152)(66,125)(67,150)(68,123)(69,148)(70,121)(71,146)(72,119)(73,144)(74,117)(75,142)(76,115)(77,140)(78,113)(79,138)(80,111)(81,136)(82,109)(83,134)(84,107)(85,132)(86,105)(87,130)(88,155)(89,128)(90,153)(91,126)(92,151)(93,124)(94,149)(95,122)(96,147)(97,120)(98,145)(99,118)(100,143)(101,116)(102,141)(103,114)(104,139)(157,181)(158,206)(159,179)(160,204)(161,177)(162,202)(163,175)(164,200)(165,173)(166,198)(167,171)(168,196)(170,194)(172,192)(174,190)(176,188)(178,186)(180,184)(183,207)(185,205)(187,203)(189,201)(191,199)(193,197) );
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)], [(1,132,208,85),(2,131,157,84),(3,130,158,83),(4,129,159,82),(5,128,160,81),(6,127,161,80),(7,126,162,79),(8,125,163,78),(9,124,164,77),(10,123,165,76),(11,122,166,75),(12,121,167,74),(13,120,168,73),(14,119,169,72),(15,118,170,71),(16,117,171,70),(17,116,172,69),(18,115,173,68),(19,114,174,67),(20,113,175,66),(21,112,176,65),(22,111,177,64),(23,110,178,63),(24,109,179,62),(25,108,180,61),(26,107,181,60),(27,106,182,59),(28,105,183,58),(29,156,184,57),(30,155,185,56),(31,154,186,55),(32,153,187,54),(33,152,188,53),(34,151,189,104),(35,150,190,103),(36,149,191,102),(37,148,192,101),(38,147,193,100),(39,146,194,99),(40,145,195,98),(41,144,196,97),(42,143,197,96),(43,142,198,95),(44,141,199,94),(45,140,200,93),(46,139,201,92),(47,138,202,91),(48,137,203,90),(49,136,204,89),(50,135,205,88),(51,134,206,87),(52,133,207,86)], [(2,26),(3,51),(4,24),(5,49),(6,22),(7,47),(8,20),(9,45),(10,18),(11,43),(12,16),(13,41),(15,39),(17,37),(19,35),(21,33),(23,31),(25,29),(28,52),(30,50),(32,48),(34,46),(36,44),(38,42),(53,112),(54,137),(55,110),(56,135),(57,108),(58,133),(59,106),(60,131),(61,156),(62,129),(63,154),(64,127),(65,152),(66,125),(67,150),(68,123),(69,148),(70,121),(71,146),(72,119),(73,144),(74,117),(75,142),(76,115),(77,140),(78,113),(79,138),(80,111),(81,136),(82,109),(83,134),(84,107),(85,132),(86,105),(87,130),(88,155),(89,128),(90,153),(91,126),(92,151),(93,124),(94,149),(95,122),(96,147),(97,120),(98,145),(99,118),(100,143),(101,116),(102,141),(103,114),(104,139),(157,181),(158,206),(159,179),(160,204),(161,177),(162,202),(163,175),(164,200),(165,173),(166,198),(167,171),(168,196),(170,194),(172,192),(174,190),(176,188),(178,186),(180,184),(183,207),(185,205),(187,203),(189,201),(191,199),(193,197)]])
74 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 13A | ··· | 13F | 26A | ··· | 26R | 26S | ··· | 26AP | 52A | ··· | 52L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 13 | ··· | 13 | 26 | ··· | 26 | 26 | ··· | 26 | 52 | ··· | 52 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 26 | 26 | 2 | 2 | 26 | 26 | 52 | 52 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
74 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | C4○D4 | D13 | D26 | D26 | C13⋊D4 | D4×D13 | D4⋊2D13 |
| kernel | C52⋊2D4 | C52⋊3C4 | C23.D13 | C2×C4×D13 | C2×C13⋊D4 | D4×C26 | C52 | D26 | C26 | C2×D4 | C2×C4 | C23 | C4 | C2 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 2 | 2 | 2 | 6 | 6 | 12 | 24 | 6 | 6 |
Matrix representation of C52⋊2D4 ►in GL6(𝔽53)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 39 | 39 | 0 | 0 |
| 0 | 0 | 6 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 29 | 45 |
| 0 | 0 | 0 | 0 | 39 | 24 |
| 39 | 46 | 0 | 0 | 0 | 0 |
| 13 | 14 | 0 | 0 | 0 | 0 |
| 0 | 0 | 45 | 11 | 0 | 0 |
| 0 | 0 | 28 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 41 | 17 |
| 0 | 0 | 0 | 0 | 29 | 12 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 49 | 52 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 42 | 0 | 0 |
| 0 | 0 | 25 | 45 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(53))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,39,6,0,0,0,0,39,40,0,0,0,0,0,0,29,39,0,0,0,0,45,24],[39,13,0,0,0,0,46,14,0,0,0,0,0,0,45,28,0,0,0,0,11,8,0,0,0,0,0,0,41,29,0,0,0,0,17,12],[1,49,0,0,0,0,0,52,0,0,0,0,0,0,8,25,0,0,0,0,42,45,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
C52⋊2D4 in GAP, Magma, Sage, TeX
C_{52}\rtimes_2D_4 % in TeX
G:=Group("C52:2D4"); // GroupNames label
G:=SmallGroup(416,159);
// by ID
G=gap.SmallGroup(416,159);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-13,103,218,188,13829]);
// Polycyclic
G:=Group<a,b,c|a^52=b^4=c^2=1,b*a*b^-1=a^-1,c*a*c=a^25,c*b*c=b^-1>;
// generators/relations