direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×D104, C26⋊1D8, C8⋊7D26, C4.7D52, C52.30D4, C104⋊8C22, D52⋊3C22, C52.29C23, C22.13D52, C13⋊1(C2×D8), (C2×C8)⋊3D13, (C2×C104)⋊5C2, (C2×D52)⋊5C2, C2.12(C2×D52), C26.10(C2×D4), (C2×C26).17D4, (C2×C4).80D26, (C2×C52).89C22, C4.27(C22×D13), SmallGroup(416,124)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×D104
G = < a,b,c | a2=b104=c2=1, ab=ba, ac=ca, cbc=b-1 >
Subgroups: 848 in 76 conjugacy classes, 33 normal (15 characteristic)
C1, C2, C2, C2, C4, C22, C22, C8, C2×C4, D4, C23, C13, C2×C8, D8, C2×D4, D13, C26, C26, C2×D8, C52, D26, C2×C26, C104, D52, D52, C2×C52, C22×D13, D104, C2×C104, C2×D52, C2×D104
Quotients: C1, C2, C22, D4, C23, D8, C2×D4, D13, C2×D8, D26, D52, C22×D13, D104, C2×D52, C2×D104
►On 208 points
Generators in S208
(1 153)(2 154)(3 155)(4 156)(5 157)(6 158)(7 159)(8 160)(9 161)(10 162)(11 163)(12 164)(13 165)(14 166)(15 167)(16 168)(17 169)(18 170)(19 171)(20 172)(21 173)(22 174)(23 175)(24 176)(25 177)(26 178)(27 179)(28 180)(29 181)(30 182)(31 183)(32 184)(33 185)(34 186)(35 187)(36 188)(37 189)(38 190)(39 191)(40 192)(41 193)(42 194)(43 195)(44 196)(45 197)(46 198)(47 199)(48 200)(49 201)(50 202)(51 203)(52 204)(53 205)(54 206)(55 207)(56 208)(57 105)(58 106)(59 107)(60 108)(61 109)(62 110)(63 111)(64 112)(65 113)(66 114)(67 115)(68 116)(69 117)(70 118)(71 119)(72 120)(73 121)(74 122)(75 123)(76 124)(77 125)(78 126)(79 127)(80 128)(81 129)(82 130)(83 131)(84 132)(85 133)(86 134)(87 135)(88 136)(89 137)(90 138)(91 139)(92 140)(93 141)(94 142)(95 143)(96 144)(97 145)(98 146)(99 147)(100 148)(101 149)(102 150)(103 151)(104 152)
(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)
(1 152)(2 151)(3 150)(4 149)(5 148)(6 147)(7 146)(8 145)(9 144)(10 143)(11 142)(12 141)(13 140)(14 139)(15 138)(16 137)(17 136)(18 135)(19 134)(20 133)(21 132)(22 131)(23 130)(24 129)(25 128)(26 127)(27 126)(28 125)(29 124)(30 123)(31 122)(32 121)(33 120)(34 119)(35 118)(36 117)(37 116)(38 115)(39 114)(40 113)(41 112)(42 111)(43 110)(44 109)(45 108)(46 107)(47 106)(48 105)(49 208)(50 207)(51 206)(52 205)(53 204)(54 203)(55 202)(56 201)(57 200)(58 199)(59 198)(60 197)(61 196)(62 195)(63 194)(64 193)(65 192)(66 191)(67 190)(68 189)(69 188)(70 187)(71 186)(72 185)(73 184)(74 183)(75 182)(76 181)(77 180)(78 179)(79 178)(80 177)(81 176)(82 175)(83 174)(84 173)(85 172)(86 171)(87 170)(88 169)(89 168)(90 167)(91 166)(92 165)(93 164)(94 163)(95 162)(96 161)(97 160)(98 159)(99 158)(100 157)(101 156)(102 155)(103 154)(104 153)
G:=sub<Sym(208)| (1,153)(2,154)(3,155)(4,156)(5,157)(6,158)(7,159)(8,160)(9,161)(10,162)(11,163)(12,164)(13,165)(14,166)(15,167)(16,168)(17,169)(18,170)(19,171)(20,172)(21,173)(22,174)(23,175)(24,176)(25,177)(26,178)(27,179)(28,180)(29,181)(30,182)(31,183)(32,184)(33,185)(34,186)(35,187)(36,188)(37,189)(38,190)(39,191)(40,192)(41,193)(42,194)(43,195)(44,196)(45,197)(46,198)(47,199)(48,200)(49,201)(50,202)(51,203)(52,204)(53,205)(54,206)(55,207)(56,208)(57,105)(58,106)(59,107)(60,108)(61,109)(62,110)(63,111)(64,112)(65,113)(66,114)(67,115)(68,116)(69,117)(70,118)(71,119)(72,120)(73,121)(74,122)(75,123)(76,124)(77,125)(78,126)(79,127)(80,128)(81,129)(82,130)(83,131)(84,132)(85,133)(86,134)(87,135)(88,136)(89,137)(90,138)(91,139)(92,140)(93,141)(94,142)(95,143)(96,144)(97,145)(98,146)(99,147)(100,148)(101,149)(102,150)(103,151)(104,152), (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), (1,152)(2,151)(3,150)(4,149)(5,148)(6,147)(7,146)(8,145)(9,144)(10,143)(11,142)(12,141)(13,140)(14,139)(15,138)(16,137)(17,136)(18,135)(19,134)(20,133)(21,132)(22,131)(23,130)(24,129)(25,128)(26,127)(27,126)(28,125)(29,124)(30,123)(31,122)(32,121)(33,120)(34,119)(35,118)(36,117)(37,116)(38,115)(39,114)(40,113)(41,112)(42,111)(43,110)(44,109)(45,108)(46,107)(47,106)(48,105)(49,208)(50,207)(51,206)(52,205)(53,204)(54,203)(55,202)(56,201)(57,200)(58,199)(59,198)(60,197)(61,196)(62,195)(63,194)(64,193)(65,192)(66,191)(67,190)(68,189)(69,188)(70,187)(71,186)(72,185)(73,184)(74,183)(75,182)(76,181)(77,180)(78,179)(79,178)(80,177)(81,176)(82,175)(83,174)(84,173)(85,172)(86,171)(87,170)(88,169)(89,168)(90,167)(91,166)(92,165)(93,164)(94,163)(95,162)(96,161)(97,160)(98,159)(99,158)(100,157)(101,156)(102,155)(103,154)(104,153)>;
G:=Group( (1,153)(2,154)(3,155)(4,156)(5,157)(6,158)(7,159)(8,160)(9,161)(10,162)(11,163)(12,164)(13,165)(14,166)(15,167)(16,168)(17,169)(18,170)(19,171)(20,172)(21,173)(22,174)(23,175)(24,176)(25,177)(26,178)(27,179)(28,180)(29,181)(30,182)(31,183)(32,184)(33,185)(34,186)(35,187)(36,188)(37,189)(38,190)(39,191)(40,192)(41,193)(42,194)(43,195)(44,196)(45,197)(46,198)(47,199)(48,200)(49,201)(50,202)(51,203)(52,204)(53,205)(54,206)(55,207)(56,208)(57,105)(58,106)(59,107)(60,108)(61,109)(62,110)(63,111)(64,112)(65,113)(66,114)(67,115)(68,116)(69,117)(70,118)(71,119)(72,120)(73,121)(74,122)(75,123)(76,124)(77,125)(78,126)(79,127)(80,128)(81,129)(82,130)(83,131)(84,132)(85,133)(86,134)(87,135)(88,136)(89,137)(90,138)(91,139)(92,140)(93,141)(94,142)(95,143)(96,144)(97,145)(98,146)(99,147)(100,148)(101,149)(102,150)(103,151)(104,152), (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), (1,152)(2,151)(3,150)(4,149)(5,148)(6,147)(7,146)(8,145)(9,144)(10,143)(11,142)(12,141)(13,140)(14,139)(15,138)(16,137)(17,136)(18,135)(19,134)(20,133)(21,132)(22,131)(23,130)(24,129)(25,128)(26,127)(27,126)(28,125)(29,124)(30,123)(31,122)(32,121)(33,120)(34,119)(35,118)(36,117)(37,116)(38,115)(39,114)(40,113)(41,112)(42,111)(43,110)(44,109)(45,108)(46,107)(47,106)(48,105)(49,208)(50,207)(51,206)(52,205)(53,204)(54,203)(55,202)(56,201)(57,200)(58,199)(59,198)(60,197)(61,196)(62,195)(63,194)(64,193)(65,192)(66,191)(67,190)(68,189)(69,188)(70,187)(71,186)(72,185)(73,184)(74,183)(75,182)(76,181)(77,180)(78,179)(79,178)(80,177)(81,176)(82,175)(83,174)(84,173)(85,172)(86,171)(87,170)(88,169)(89,168)(90,167)(91,166)(92,165)(93,164)(94,163)(95,162)(96,161)(97,160)(98,159)(99,158)(100,157)(101,156)(102,155)(103,154)(104,153) );
G=PermutationGroup([[(1,153),(2,154),(3,155),(4,156),(5,157),(6,158),(7,159),(8,160),(9,161),(10,162),(11,163),(12,164),(13,165),(14,166),(15,167),(16,168),(17,169),(18,170),(19,171),(20,172),(21,173),(22,174),(23,175),(24,176),(25,177),(26,178),(27,179),(28,180),(29,181),(30,182),(31,183),(32,184),(33,185),(34,186),(35,187),(36,188),(37,189),(38,190),(39,191),(40,192),(41,193),(42,194),(43,195),(44,196),(45,197),(46,198),(47,199),(48,200),(49,201),(50,202),(51,203),(52,204),(53,205),(54,206),(55,207),(56,208),(57,105),(58,106),(59,107),(60,108),(61,109),(62,110),(63,111),(64,112),(65,113),(66,114),(67,115),(68,116),(69,117),(70,118),(71,119),(72,120),(73,121),(74,122),(75,123),(76,124),(77,125),(78,126),(79,127),(80,128),(81,129),(82,130),(83,131),(84,132),(85,133),(86,134),(87,135),(88,136),(89,137),(90,138),(91,139),(92,140),(93,141),(94,142),(95,143),(96,144),(97,145),(98,146),(99,147),(100,148),(101,149),(102,150),(103,151),(104,152)], [(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)], [(1,152),(2,151),(3,150),(4,149),(5,148),(6,147),(7,146),(8,145),(9,144),(10,143),(11,142),(12,141),(13,140),(14,139),(15,138),(16,137),(17,136),(18,135),(19,134),(20,133),(21,132),(22,131),(23,130),(24,129),(25,128),(26,127),(27,126),(28,125),(29,124),(30,123),(31,122),(32,121),(33,120),(34,119),(35,118),(36,117),(37,116),(38,115),(39,114),(40,113),(41,112),(42,111),(43,110),(44,109),(45,108),(46,107),(47,106),(48,105),(49,208),(50,207),(51,206),(52,205),(53,204),(54,203),(55,202),(56,201),(57,200),(58,199),(59,198),(60,197),(61,196),(62,195),(63,194),(64,193),(65,192),(66,191),(67,190),(68,189),(69,188),(70,187),(71,186),(72,185),(73,184),(74,183),(75,182),(76,181),(77,180),(78,179),(79,178),(80,177),(81,176),(82,175),(83,174),(84,173),(85,172),(86,171),(87,170),(88,169),(89,168),(90,167),(91,166),(92,165),(93,164),(94,163),(95,162),(96,161),(97,160),(98,159),(99,158),(100,157),(101,156),(102,155),(103,154),(104,153)]])
110 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 8A | 8B | 8C | 8D | 13A | ··· | 13F | 26A | ··· | 26R | 52A | ··· | 52X | 104A | ··· | 104AV |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 8 | 8 | 8 | 8 | 13 | ··· | 13 | 26 | ··· | 26 | 52 | ··· | 52 | 104 | ··· | 104 |
| size | 1 | 1 | 1 | 1 | 52 | 52 | 52 | 52 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
110 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | D4 | D4 | D8 | D13 | D26 | D26 | D52 | D52 | D104 |
| kernel | C2×D104 | D104 | C2×C104 | C2×D52 | C52 | C2×C26 | C26 | C2×C8 | C8 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 4 | 1 | 2 | 1 | 1 | 4 | 6 | 12 | 6 | 12 | 12 | 48 |
Matrix representation of C2×D104 ►in GL3(𝔽313) generated by
| 312 | 0 | 0 |
| 0 | 312 | 0 |
| 0 | 0 | 312 |
| 312 | 0 | 0 |
| 0 | 92 | 4 |
| 0 | 2 | 160 |
| 312 | 0 | 0 |
| 0 | 65 | 163 |
| 0 | 241 | 248 |
G:=sub<GL(3,GF(313))| [312,0,0,0,312,0,0,0,312],[312,0,0,0,92,2,0,4,160],[312,0,0,0,65,241,0,163,248] >;
C2×D104 in GAP, Magma, Sage, TeX
C_2\times D_{104} % in TeX
G:=Group("C2xD104"); // GroupNames label
G:=SmallGroup(416,124);
// by ID
G=gap.SmallGroup(416,124);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-13,218,122,579,69,13829]);
// Polycyclic
G:=Group<a,b,c|a^2=b^104=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations