metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4⋊1D52, C52⋊4D4, C42⋊5D13, (C4×C52)⋊4C2, (C2×D52)⋊1C2, C26.3(C2×D4), C2.5(C2×D52), C13⋊1(C4⋊1D4), (C2×C4).76D26, (C2×C26).15C23, (C2×C52).87C22, (C22×D13).1C22, C22.36(C22×D13), SmallGroup(416,95)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4⋊D52
G = < a,b,c | a4=b52=c2=1, ab=ba, cac=a-1, cbc=b-1 >
Subgroups: 1080 in 108 conjugacy classes, 41 normal (7 characteristic)
C1, C2, C2, C4, C22, C22, C2×C4, D4, C23, C13, C42, C2×D4, D13, C26, C4⋊1D4, C52, D26, C2×C26, D52, C2×C52, C22×D13, C4×C52, C2×D52, C4⋊D52
Quotients: C1, C2, C22, D4, C23, C2×D4, D13, C4⋊1D4, D26, D52, C22×D13, C2×D52, C4⋊D52
►On 208 points
Generators in S208
(1 64 121 182)(2 65 122 183)(3 66 123 184)(4 67 124 185)(5 68 125 186)(6 69 126 187)(7 70 127 188)(8 71 128 189)(9 72 129 190)(10 73 130 191)(11 74 131 192)(12 75 132 193)(13 76 133 194)(14 77 134 195)(15 78 135 196)(16 79 136 197)(17 80 137 198)(18 81 138 199)(19 82 139 200)(20 83 140 201)(21 84 141 202)(22 85 142 203)(23 86 143 204)(24 87 144 205)(25 88 145 206)(26 89 146 207)(27 90 147 208)(28 91 148 157)(29 92 149 158)(30 93 150 159)(31 94 151 160)(32 95 152 161)(33 96 153 162)(34 97 154 163)(35 98 155 164)(36 99 156 165)(37 100 105 166)(38 101 106 167)(39 102 107 168)(40 103 108 169)(41 104 109 170)(42 53 110 171)(43 54 111 172)(44 55 112 173)(45 56 113 174)(46 57 114 175)(47 58 115 176)(48 59 116 177)(49 60 117 178)(50 61 118 179)(51 62 119 180)(52 63 120 181)
(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)
(1 168)(2 167)(3 166)(4 165)(5 164)(6 163)(7 162)(8 161)(9 160)(10 159)(11 158)(12 157)(13 208)(14 207)(15 206)(16 205)(17 204)(18 203)(19 202)(20 201)(21 200)(22 199)(23 198)(24 197)(25 196)(26 195)(27 194)(28 193)(29 192)(30 191)(31 190)(32 189)(33 188)(34 187)(35 186)(36 185)(37 184)(38 183)(39 182)(40 181)(41 180)(42 179)(43 178)(44 177)(45 176)(46 175)(47 174)(48 173)(49 172)(50 171)(51 170)(52 169)(53 118)(54 117)(55 116)(56 115)(57 114)(58 113)(59 112)(60 111)(61 110)(62 109)(63 108)(64 107)(65 106)(66 105)(67 156)(68 155)(69 154)(70 153)(71 152)(72 151)(73 150)(74 149)(75 148)(76 147)(77 146)(78 145)(79 144)(80 143)(81 142)(82 141)(83 140)(84 139)(85 138)(86 137)(87 136)(88 135)(89 134)(90 133)(91 132)(92 131)(93 130)(94 129)(95 128)(96 127)(97 126)(98 125)(99 124)(100 123)(101 122)(102 121)(103 120)(104 119)
G:=sub<Sym(208)| (1,64,121,182)(2,65,122,183)(3,66,123,184)(4,67,124,185)(5,68,125,186)(6,69,126,187)(7,70,127,188)(8,71,128,189)(9,72,129,190)(10,73,130,191)(11,74,131,192)(12,75,132,193)(13,76,133,194)(14,77,134,195)(15,78,135,196)(16,79,136,197)(17,80,137,198)(18,81,138,199)(19,82,139,200)(20,83,140,201)(21,84,141,202)(22,85,142,203)(23,86,143,204)(24,87,144,205)(25,88,145,206)(26,89,146,207)(27,90,147,208)(28,91,148,157)(29,92,149,158)(30,93,150,159)(31,94,151,160)(32,95,152,161)(33,96,153,162)(34,97,154,163)(35,98,155,164)(36,99,156,165)(37,100,105,166)(38,101,106,167)(39,102,107,168)(40,103,108,169)(41,104,109,170)(42,53,110,171)(43,54,111,172)(44,55,112,173)(45,56,113,174)(46,57,114,175)(47,58,115,176)(48,59,116,177)(49,60,117,178)(50,61,118,179)(51,62,119,180)(52,63,120,181), (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), (1,168)(2,167)(3,166)(4,165)(5,164)(6,163)(7,162)(8,161)(9,160)(10,159)(11,158)(12,157)(13,208)(14,207)(15,206)(16,205)(17,204)(18,203)(19,202)(20,201)(21,200)(22,199)(23,198)(24,197)(25,196)(26,195)(27,194)(28,193)(29,192)(30,191)(31,190)(32,189)(33,188)(34,187)(35,186)(36,185)(37,184)(38,183)(39,182)(40,181)(41,180)(42,179)(43,178)(44,177)(45,176)(46,175)(47,174)(48,173)(49,172)(50,171)(51,170)(52,169)(53,118)(54,117)(55,116)(56,115)(57,114)(58,113)(59,112)(60,111)(61,110)(62,109)(63,108)(64,107)(65,106)(66,105)(67,156)(68,155)(69,154)(70,153)(71,152)(72,151)(73,150)(74,149)(75,148)(76,147)(77,146)(78,145)(79,144)(80,143)(81,142)(82,141)(83,140)(84,139)(85,138)(86,137)(87,136)(88,135)(89,134)(90,133)(91,132)(92,131)(93,130)(94,129)(95,128)(96,127)(97,126)(98,125)(99,124)(100,123)(101,122)(102,121)(103,120)(104,119)>;
G:=Group( (1,64,121,182)(2,65,122,183)(3,66,123,184)(4,67,124,185)(5,68,125,186)(6,69,126,187)(7,70,127,188)(8,71,128,189)(9,72,129,190)(10,73,130,191)(11,74,131,192)(12,75,132,193)(13,76,133,194)(14,77,134,195)(15,78,135,196)(16,79,136,197)(17,80,137,198)(18,81,138,199)(19,82,139,200)(20,83,140,201)(21,84,141,202)(22,85,142,203)(23,86,143,204)(24,87,144,205)(25,88,145,206)(26,89,146,207)(27,90,147,208)(28,91,148,157)(29,92,149,158)(30,93,150,159)(31,94,151,160)(32,95,152,161)(33,96,153,162)(34,97,154,163)(35,98,155,164)(36,99,156,165)(37,100,105,166)(38,101,106,167)(39,102,107,168)(40,103,108,169)(41,104,109,170)(42,53,110,171)(43,54,111,172)(44,55,112,173)(45,56,113,174)(46,57,114,175)(47,58,115,176)(48,59,116,177)(49,60,117,178)(50,61,118,179)(51,62,119,180)(52,63,120,181), (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), (1,168)(2,167)(3,166)(4,165)(5,164)(6,163)(7,162)(8,161)(9,160)(10,159)(11,158)(12,157)(13,208)(14,207)(15,206)(16,205)(17,204)(18,203)(19,202)(20,201)(21,200)(22,199)(23,198)(24,197)(25,196)(26,195)(27,194)(28,193)(29,192)(30,191)(31,190)(32,189)(33,188)(34,187)(35,186)(36,185)(37,184)(38,183)(39,182)(40,181)(41,180)(42,179)(43,178)(44,177)(45,176)(46,175)(47,174)(48,173)(49,172)(50,171)(51,170)(52,169)(53,118)(54,117)(55,116)(56,115)(57,114)(58,113)(59,112)(60,111)(61,110)(62,109)(63,108)(64,107)(65,106)(66,105)(67,156)(68,155)(69,154)(70,153)(71,152)(72,151)(73,150)(74,149)(75,148)(76,147)(77,146)(78,145)(79,144)(80,143)(81,142)(82,141)(83,140)(84,139)(85,138)(86,137)(87,136)(88,135)(89,134)(90,133)(91,132)(92,131)(93,130)(94,129)(95,128)(96,127)(97,126)(98,125)(99,124)(100,123)(101,122)(102,121)(103,120)(104,119) );
G=PermutationGroup([[(1,64,121,182),(2,65,122,183),(3,66,123,184),(4,67,124,185),(5,68,125,186),(6,69,126,187),(7,70,127,188),(8,71,128,189),(9,72,129,190),(10,73,130,191),(11,74,131,192),(12,75,132,193),(13,76,133,194),(14,77,134,195),(15,78,135,196),(16,79,136,197),(17,80,137,198),(18,81,138,199),(19,82,139,200),(20,83,140,201),(21,84,141,202),(22,85,142,203),(23,86,143,204),(24,87,144,205),(25,88,145,206),(26,89,146,207),(27,90,147,208),(28,91,148,157),(29,92,149,158),(30,93,150,159),(31,94,151,160),(32,95,152,161),(33,96,153,162),(34,97,154,163),(35,98,155,164),(36,99,156,165),(37,100,105,166),(38,101,106,167),(39,102,107,168),(40,103,108,169),(41,104,109,170),(42,53,110,171),(43,54,111,172),(44,55,112,173),(45,56,113,174),(46,57,114,175),(47,58,115,176),(48,59,116,177),(49,60,117,178),(50,61,118,179),(51,62,119,180),(52,63,120,181)], [(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)], [(1,168),(2,167),(3,166),(4,165),(5,164),(6,163),(7,162),(8,161),(9,160),(10,159),(11,158),(12,157),(13,208),(14,207),(15,206),(16,205),(17,204),(18,203),(19,202),(20,201),(21,200),(22,199),(23,198),(24,197),(25,196),(26,195),(27,194),(28,193),(29,192),(30,191),(31,190),(32,189),(33,188),(34,187),(35,186),(36,185),(37,184),(38,183),(39,182),(40,181),(41,180),(42,179),(43,178),(44,177),(45,176),(46,175),(47,174),(48,173),(49,172),(50,171),(51,170),(52,169),(53,118),(54,117),(55,116),(56,115),(57,114),(58,113),(59,112),(60,111),(61,110),(62,109),(63,108),(64,107),(65,106),(66,105),(67,156),(68,155),(69,154),(70,153),(71,152),(72,151),(73,150),(74,149),(75,148),(76,147),(77,146),(78,145),(79,144),(80,143),(81,142),(82,141),(83,140),(84,139),(85,138),(86,137),(87,136),(88,135),(89,134),(90,133),(91,132),(92,131),(93,130),(94,129),(95,128),(96,127),(97,126),(98,125),(99,124),(100,123),(101,122),(102,121),(103,120),(104,119)]])
110 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | ··· | 4F | 13A | ··· | 13F | 26A | ··· | 26R | 52A | ··· | 52BT |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 13 | ··· | 13 | 26 | ··· | 26 | 52 | ··· | 52 |
| size | 1 | 1 | 1 | 1 | 52 | 52 | 52 | 52 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
110 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | D4 | D13 | D26 | D52 |
| kernel | C4⋊D52 | C4×C52 | C2×D52 | C52 | C42 | C2×C4 | C4 |
| # reps | 1 | 1 | 6 | 6 | 6 | 18 | 72 |
Matrix representation of C4⋊D52 ►in GL4(𝔽53) generated by
| 34 | 16 | 0 | 0 |
| 37 | 19 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 4 | 51 | 0 | 0 |
| 2 | 39 | 0 | 0 |
| 0 | 0 | 4 | 51 |
| 0 | 0 | 2 | 39 |
| 49 | 2 | 0 | 0 |
| 19 | 4 | 0 | 0 |
| 0 | 0 | 9 | 26 |
| 0 | 0 | 1 | 44 |
G:=sub<GL(4,GF(53))| [34,37,0,0,16,19,0,0,0,0,1,0,0,0,0,1],[4,2,0,0,51,39,0,0,0,0,4,2,0,0,51,39],[49,19,0,0,2,4,0,0,0,0,9,1,0,0,26,44] >;
C4⋊D52 in GAP, Magma, Sage, TeX
C_4\rtimes D_{52} % in TeX
G:=Group("C4:D52"); // GroupNames label
G:=SmallGroup(416,95);
// by ID
G=gap.SmallGroup(416,95);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-13,217,103,218,50,13829]);
// Polycyclic
G:=Group<a,b,c|a^4=b^52=c^2=1,a*b=b*a,c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations