direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C52.4C4, C26⋊4M4(2), C52.41C23, C23.3Dic13, C52.57(C2×C4), (C2×C52).20C4, C13⋊6(C2×M4(2)), (C2×C4).100D26, (C22×C52).9C2, C4.9(C2×Dic13), (C2×C4).6Dic13, (C22×C4).4D13, C13⋊2C8⋊12C22, (C22×C26).11C4, C26.34(C22×C4), C4.41(C22×D13), (C2×C52).100C22, C2.3(C22×Dic13), C22.12(C2×Dic13), (C2×C13⋊2C8)⋊12C2, (C2×C26).52(C2×C4), SmallGroup(416,142)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C52.4C4
G = < a,b,c | a2=b52=1, c4=b26, ab=ba, ac=ca, cbc-1=b-1 >
Subgroups: 208 in 68 conjugacy classes, 49 normal (19 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C23, C13, C2×C8, M4(2), C22×C4, C26, C26, C26, C2×M4(2), C52, C52, C2×C26, C2×C26, C2×C26, C13⋊2C8, C2×C52, C2×C52, C22×C26, C2×C13⋊2C8, C52.4C4, C22×C52, C2×C52.4C4
Quotients: C1, C2, C4, C22, C2×C4, C23, M4(2), C22×C4, D13, C2×M4(2), Dic13, D26, C2×Dic13, C22×D13, C52.4C4, C22×Dic13, C2×C52.4C4
►On 208 points
Generators in S208
(1 127)(2 128)(3 129)(4 130)(5 131)(6 132)(7 133)(8 134)(9 135)(10 136)(11 137)(12 138)(13 139)(14 140)(15 141)(16 142)(17 143)(18 144)(19 145)(20 146)(21 147)(22 148)(23 149)(24 150)(25 151)(26 152)(27 153)(28 154)(29 155)(30 156)(31 105)(32 106)(33 107)(34 108)(35 109)(36 110)(37 111)(38 112)(39 113)(40 114)(41 115)(42 116)(43 117)(44 118)(45 119)(46 120)(47 121)(48 122)(49 123)(50 124)(51 125)(52 126)(53 192)(54 193)(55 194)(56 195)(57 196)(58 197)(59 198)(60 199)(61 200)(62 201)(63 202)(64 203)(65 204)(66 205)(67 206)(68 207)(69 208)(70 157)(71 158)(72 159)(73 160)(74 161)(75 162)(76 163)(77 164)(78 165)(79 166)(80 167)(81 168)(82 169)(83 170)(84 171)(85 172)(86 173)(87 174)(88 175)(89 176)(90 177)(91 178)(92 179)(93 180)(94 181)(95 182)(96 183)(97 184)(98 185)(99 186)(100 187)(101 188)(102 189)(103 190)(104 191)
(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 54 140 180 27 80 114 206)(2 53 141 179 28 79 115 205)(3 104 142 178 29 78 116 204)(4 103 143 177 30 77 117 203)(5 102 144 176 31 76 118 202)(6 101 145 175 32 75 119 201)(7 100 146 174 33 74 120 200)(8 99 147 173 34 73 121 199)(9 98 148 172 35 72 122 198)(10 97 149 171 36 71 123 197)(11 96 150 170 37 70 124 196)(12 95 151 169 38 69 125 195)(13 94 152 168 39 68 126 194)(14 93 153 167 40 67 127 193)(15 92 154 166 41 66 128 192)(16 91 155 165 42 65 129 191)(17 90 156 164 43 64 130 190)(18 89 105 163 44 63 131 189)(19 88 106 162 45 62 132 188)(20 87 107 161 46 61 133 187)(21 86 108 160 47 60 134 186)(22 85 109 159 48 59 135 185)(23 84 110 158 49 58 136 184)(24 83 111 157 50 57 137 183)(25 82 112 208 51 56 138 182)(26 81 113 207 52 55 139 181)
G:=sub<Sym(208)| (1,127)(2,128)(3,129)(4,130)(5,131)(6,132)(7,133)(8,134)(9,135)(10,136)(11,137)(12,138)(13,139)(14,140)(15,141)(16,142)(17,143)(18,144)(19,145)(20,146)(21,147)(22,148)(23,149)(24,150)(25,151)(26,152)(27,153)(28,154)(29,155)(30,156)(31,105)(32,106)(33,107)(34,108)(35,109)(36,110)(37,111)(38,112)(39,113)(40,114)(41,115)(42,116)(43,117)(44,118)(45,119)(46,120)(47,121)(48,122)(49,123)(50,124)(51,125)(52,126)(53,192)(54,193)(55,194)(56,195)(57,196)(58,197)(59,198)(60,199)(61,200)(62,201)(63,202)(64,203)(65,204)(66,205)(67,206)(68,207)(69,208)(70,157)(71,158)(72,159)(73,160)(74,161)(75,162)(76,163)(77,164)(78,165)(79,166)(80,167)(81,168)(82,169)(83,170)(84,171)(85,172)(86,173)(87,174)(88,175)(89,176)(90,177)(91,178)(92,179)(93,180)(94,181)(95,182)(96,183)(97,184)(98,185)(99,186)(100,187)(101,188)(102,189)(103,190)(104,191), (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), (1,54,140,180,27,80,114,206)(2,53,141,179,28,79,115,205)(3,104,142,178,29,78,116,204)(4,103,143,177,30,77,117,203)(5,102,144,176,31,76,118,202)(6,101,145,175,32,75,119,201)(7,100,146,174,33,74,120,200)(8,99,147,173,34,73,121,199)(9,98,148,172,35,72,122,198)(10,97,149,171,36,71,123,197)(11,96,150,170,37,70,124,196)(12,95,151,169,38,69,125,195)(13,94,152,168,39,68,126,194)(14,93,153,167,40,67,127,193)(15,92,154,166,41,66,128,192)(16,91,155,165,42,65,129,191)(17,90,156,164,43,64,130,190)(18,89,105,163,44,63,131,189)(19,88,106,162,45,62,132,188)(20,87,107,161,46,61,133,187)(21,86,108,160,47,60,134,186)(22,85,109,159,48,59,135,185)(23,84,110,158,49,58,136,184)(24,83,111,157,50,57,137,183)(25,82,112,208,51,56,138,182)(26,81,113,207,52,55,139,181)>;
G:=Group( (1,127)(2,128)(3,129)(4,130)(5,131)(6,132)(7,133)(8,134)(9,135)(10,136)(11,137)(12,138)(13,139)(14,140)(15,141)(16,142)(17,143)(18,144)(19,145)(20,146)(21,147)(22,148)(23,149)(24,150)(25,151)(26,152)(27,153)(28,154)(29,155)(30,156)(31,105)(32,106)(33,107)(34,108)(35,109)(36,110)(37,111)(38,112)(39,113)(40,114)(41,115)(42,116)(43,117)(44,118)(45,119)(46,120)(47,121)(48,122)(49,123)(50,124)(51,125)(52,126)(53,192)(54,193)(55,194)(56,195)(57,196)(58,197)(59,198)(60,199)(61,200)(62,201)(63,202)(64,203)(65,204)(66,205)(67,206)(68,207)(69,208)(70,157)(71,158)(72,159)(73,160)(74,161)(75,162)(76,163)(77,164)(78,165)(79,166)(80,167)(81,168)(82,169)(83,170)(84,171)(85,172)(86,173)(87,174)(88,175)(89,176)(90,177)(91,178)(92,179)(93,180)(94,181)(95,182)(96,183)(97,184)(98,185)(99,186)(100,187)(101,188)(102,189)(103,190)(104,191), (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), (1,54,140,180,27,80,114,206)(2,53,141,179,28,79,115,205)(3,104,142,178,29,78,116,204)(4,103,143,177,30,77,117,203)(5,102,144,176,31,76,118,202)(6,101,145,175,32,75,119,201)(7,100,146,174,33,74,120,200)(8,99,147,173,34,73,121,199)(9,98,148,172,35,72,122,198)(10,97,149,171,36,71,123,197)(11,96,150,170,37,70,124,196)(12,95,151,169,38,69,125,195)(13,94,152,168,39,68,126,194)(14,93,153,167,40,67,127,193)(15,92,154,166,41,66,128,192)(16,91,155,165,42,65,129,191)(17,90,156,164,43,64,130,190)(18,89,105,163,44,63,131,189)(19,88,106,162,45,62,132,188)(20,87,107,161,46,61,133,187)(21,86,108,160,47,60,134,186)(22,85,109,159,48,59,135,185)(23,84,110,158,49,58,136,184)(24,83,111,157,50,57,137,183)(25,82,112,208,51,56,138,182)(26,81,113,207,52,55,139,181) );
G=PermutationGroup([[(1,127),(2,128),(3,129),(4,130),(5,131),(6,132),(7,133),(8,134),(9,135),(10,136),(11,137),(12,138),(13,139),(14,140),(15,141),(16,142),(17,143),(18,144),(19,145),(20,146),(21,147),(22,148),(23,149),(24,150),(25,151),(26,152),(27,153),(28,154),(29,155),(30,156),(31,105),(32,106),(33,107),(34,108),(35,109),(36,110),(37,111),(38,112),(39,113),(40,114),(41,115),(42,116),(43,117),(44,118),(45,119),(46,120),(47,121),(48,122),(49,123),(50,124),(51,125),(52,126),(53,192),(54,193),(55,194),(56,195),(57,196),(58,197),(59,198),(60,199),(61,200),(62,201),(63,202),(64,203),(65,204),(66,205),(67,206),(68,207),(69,208),(70,157),(71,158),(72,159),(73,160),(74,161),(75,162),(76,163),(77,164),(78,165),(79,166),(80,167),(81,168),(82,169),(83,170),(84,171),(85,172),(86,173),(87,174),(88,175),(89,176),(90,177),(91,178),(92,179),(93,180),(94,181),(95,182),(96,183),(97,184),(98,185),(99,186),(100,187),(101,188),(102,189),(103,190),(104,191)], [(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)], [(1,54,140,180,27,80,114,206),(2,53,141,179,28,79,115,205),(3,104,142,178,29,78,116,204),(4,103,143,177,30,77,117,203),(5,102,144,176,31,76,118,202),(6,101,145,175,32,75,119,201),(7,100,146,174,33,74,120,200),(8,99,147,173,34,73,121,199),(9,98,148,172,35,72,122,198),(10,97,149,171,36,71,123,197),(11,96,150,170,37,70,124,196),(12,95,151,169,38,69,125,195),(13,94,152,168,39,68,126,194),(14,93,153,167,40,67,127,193),(15,92,154,166,41,66,128,192),(16,91,155,165,42,65,129,191),(17,90,156,164,43,64,130,190),(18,89,105,163,44,63,131,189),(19,88,106,162,45,62,132,188),(20,87,107,161,46,61,133,187),(21,86,108,160,47,60,134,186),(22,85,109,159,48,59,135,185),(23,84,110,158,49,58,136,184),(24,83,111,157,50,57,137,183),(25,82,112,208,51,56,138,182),(26,81,113,207,52,55,139,181)]])
116 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 8A | ··· | 8H | 13A | ··· | 13F | 26A | ··· | 26AP | 52A | ··· | 52AV |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 13 | ··· | 13 | 26 | ··· | 26 | 52 | ··· | 52 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 26 | ··· | 26 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
116 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | - | ||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | M4(2) | D13 | Dic13 | D26 | Dic13 | C52.4C4 |
| kernel | C2×C52.4C4 | C2×C13⋊2C8 | C52.4C4 | C22×C52 | C2×C52 | C22×C26 | C26 | C22×C4 | C2×C4 | C2×C4 | C23 | C2 |
| # reps | 1 | 2 | 4 | 1 | 6 | 2 | 4 | 6 | 18 | 18 | 6 | 48 |
Matrix representation of C2×C52.4C4 ►in GL3(𝔽313) generated by
| 312 | 0 | 0 |
| 0 | 312 | 0 |
| 0 | 0 | 312 |
| 312 | 0 | 0 |
| 0 | 151 | 0 |
| 0 | 0 | 199 |
| 25 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 288 | 0 |
G:=sub<GL(3,GF(313))| [312,0,0,0,312,0,0,0,312],[312,0,0,0,151,0,0,0,199],[25,0,0,0,0,288,0,1,0] >;
C2×C52.4C4 in GAP, Magma, Sage, TeX
C_2\times C_{52}._4C_4 % in TeX
G:=Group("C2xC52.4C4"); // GroupNames label
G:=SmallGroup(416,142);
// by ID
G=gap.SmallGroup(416,142);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-13,48,362,69,13829]);
// Polycyclic
G:=Group<a,b,c|a^2=b^52=1,c^4=b^26,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations