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 [×2], C2 [×4], C4 [×2], C22, C22 [×8], C8 [×2], C2×C4, D4 [×6], C23 [×2], C13, C2×C8, D8 [×4], C2×D4 [×2], D13 [×4], C26, C26 [×2], C2×D8, C52 [×2], D26 [×8], C2×C26, C104 [×2], D52 [×4], D52 [×2], C2×C52, C22×D13 [×2], D104 [×4], C2×C104, C2×D52 [×2], C2×D104
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D8 [×2], C2×D4, D13, C2×D8, D26 [×3], D52 [×2], C22×D13, D104 [×2], C2×D52, C2×D104
►On 208 points
Generators in S208
(1 128)(2 129)(3 130)(4 131)(5 132)(6 133)(7 134)(8 135)(9 136)(10 137)(11 138)(12 139)(13 140)(14 141)(15 142)(16 143)(17 144)(18 145)(19 146)(20 147)(21 148)(22 149)(23 150)(24 151)(25 152)(26 153)(27 154)(28 155)(29 156)(30 157)(31 158)(32 159)(33 160)(34 161)(35 162)(36 163)(37 164)(38 165)(39 166)(40 167)(41 168)(42 169)(43 170)(44 171)(45 172)(46 173)(47 174)(48 175)(49 176)(50 177)(51 178)(52 179)(53 180)(54 181)(55 182)(56 183)(57 184)(58 185)(59 186)(60 187)(61 188)(62 189)(63 190)(64 191)(65 192)(66 193)(67 194)(68 195)(69 196)(70 197)(71 198)(72 199)(73 200)(74 201)(75 202)(76 203)(77 204)(78 205)(79 206)(80 207)(81 208)(82 105)(83 106)(84 107)(85 108)(86 109)(87 110)(88 111)(89 112)(90 113)(91 114)(92 115)(93 116)(94 117)(95 118)(96 119)(97 120)(98 121)(99 122)(100 123)(101 124)(102 125)(103 126)(104 127)
(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)
(1 127)(2 126)(3 125)(4 124)(5 123)(6 122)(7 121)(8 120)(9 119)(10 118)(11 117)(12 116)(13 115)(14 114)(15 113)(16 112)(17 111)(18 110)(19 109)(20 108)(21 107)(22 106)(23 105)(24 208)(25 207)(26 206)(27 205)(28 204)(29 203)(30 202)(31 201)(32 200)(33 199)(34 198)(35 197)(36 196)(37 195)(38 194)(39 193)(40 192)(41 191)(42 190)(43 189)(44 188)(45 187)(46 186)(47 185)(48 184)(49 183)(50 182)(51 181)(52 180)(53 179)(54 178)(55 177)(56 176)(57 175)(58 174)(59 173)(60 172)(61 171)(62 170)(63 169)(64 168)(65 167)(66 166)(67 165)(68 164)(69 163)(70 162)(71 161)(72 160)(73 159)(74 158)(75 157)(76 156)(77 155)(78 154)(79 153)(80 152)(81 151)(82 150)(83 149)(84 148)(85 147)(86 146)(87 145)(88 144)(89 143)(90 142)(91 141)(92 140)(93 139)(94 138)(95 137)(96 136)(97 135)(98 134)(99 133)(100 132)(101 131)(102 130)(103 129)(104 128)
G:=sub<Sym(208)| (1,128)(2,129)(3,130)(4,131)(5,132)(6,133)(7,134)(8,135)(9,136)(10,137)(11,138)(12,139)(13,140)(14,141)(15,142)(16,143)(17,144)(18,145)(19,146)(20,147)(21,148)(22,149)(23,150)(24,151)(25,152)(26,153)(27,154)(28,155)(29,156)(30,157)(31,158)(32,159)(33,160)(34,161)(35,162)(36,163)(37,164)(38,165)(39,166)(40,167)(41,168)(42,169)(43,170)(44,171)(45,172)(46,173)(47,174)(48,175)(49,176)(50,177)(51,178)(52,179)(53,180)(54,181)(55,182)(56,183)(57,184)(58,185)(59,186)(60,187)(61,188)(62,189)(63,190)(64,191)(65,192)(66,193)(67,194)(68,195)(69,196)(70,197)(71,198)(72,199)(73,200)(74,201)(75,202)(76,203)(77,204)(78,205)(79,206)(80,207)(81,208)(82,105)(83,106)(84,107)(85,108)(86,109)(87,110)(88,111)(89,112)(90,113)(91,114)(92,115)(93,116)(94,117)(95,118)(96,119)(97,120)(98,121)(99,122)(100,123)(101,124)(102,125)(103,126)(104,127), (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), (1,127)(2,126)(3,125)(4,124)(5,123)(6,122)(7,121)(8,120)(9,119)(10,118)(11,117)(12,116)(13,115)(14,114)(15,113)(16,112)(17,111)(18,110)(19,109)(20,108)(21,107)(22,106)(23,105)(24,208)(25,207)(26,206)(27,205)(28,204)(29,203)(30,202)(31,201)(32,200)(33,199)(34,198)(35,197)(36,196)(37,195)(38,194)(39,193)(40,192)(41,191)(42,190)(43,189)(44,188)(45,187)(46,186)(47,185)(48,184)(49,183)(50,182)(51,181)(52,180)(53,179)(54,178)(55,177)(56,176)(57,175)(58,174)(59,173)(60,172)(61,171)(62,170)(63,169)(64,168)(65,167)(66,166)(67,165)(68,164)(69,163)(70,162)(71,161)(72,160)(73,159)(74,158)(75,157)(76,156)(77,155)(78,154)(79,153)(80,152)(81,151)(82,150)(83,149)(84,148)(85,147)(86,146)(87,145)(88,144)(89,143)(90,142)(91,141)(92,140)(93,139)(94,138)(95,137)(96,136)(97,135)(98,134)(99,133)(100,132)(101,131)(102,130)(103,129)(104,128)>;
G:=Group( (1,128)(2,129)(3,130)(4,131)(5,132)(6,133)(7,134)(8,135)(9,136)(10,137)(11,138)(12,139)(13,140)(14,141)(15,142)(16,143)(17,144)(18,145)(19,146)(20,147)(21,148)(22,149)(23,150)(24,151)(25,152)(26,153)(27,154)(28,155)(29,156)(30,157)(31,158)(32,159)(33,160)(34,161)(35,162)(36,163)(37,164)(38,165)(39,166)(40,167)(41,168)(42,169)(43,170)(44,171)(45,172)(46,173)(47,174)(48,175)(49,176)(50,177)(51,178)(52,179)(53,180)(54,181)(55,182)(56,183)(57,184)(58,185)(59,186)(60,187)(61,188)(62,189)(63,190)(64,191)(65,192)(66,193)(67,194)(68,195)(69,196)(70,197)(71,198)(72,199)(73,200)(74,201)(75,202)(76,203)(77,204)(78,205)(79,206)(80,207)(81,208)(82,105)(83,106)(84,107)(85,108)(86,109)(87,110)(88,111)(89,112)(90,113)(91,114)(92,115)(93,116)(94,117)(95,118)(96,119)(97,120)(98,121)(99,122)(100,123)(101,124)(102,125)(103,126)(104,127), (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), (1,127)(2,126)(3,125)(4,124)(5,123)(6,122)(7,121)(8,120)(9,119)(10,118)(11,117)(12,116)(13,115)(14,114)(15,113)(16,112)(17,111)(18,110)(19,109)(20,108)(21,107)(22,106)(23,105)(24,208)(25,207)(26,206)(27,205)(28,204)(29,203)(30,202)(31,201)(32,200)(33,199)(34,198)(35,197)(36,196)(37,195)(38,194)(39,193)(40,192)(41,191)(42,190)(43,189)(44,188)(45,187)(46,186)(47,185)(48,184)(49,183)(50,182)(51,181)(52,180)(53,179)(54,178)(55,177)(56,176)(57,175)(58,174)(59,173)(60,172)(61,171)(62,170)(63,169)(64,168)(65,167)(66,166)(67,165)(68,164)(69,163)(70,162)(71,161)(72,160)(73,159)(74,158)(75,157)(76,156)(77,155)(78,154)(79,153)(80,152)(81,151)(82,150)(83,149)(84,148)(85,147)(86,146)(87,145)(88,144)(89,143)(90,142)(91,141)(92,140)(93,139)(94,138)(95,137)(96,136)(97,135)(98,134)(99,133)(100,132)(101,131)(102,130)(103,129)(104,128) );
G=PermutationGroup([(1,128),(2,129),(3,130),(4,131),(5,132),(6,133),(7,134),(8,135),(9,136),(10,137),(11,138),(12,139),(13,140),(14,141),(15,142),(16,143),(17,144),(18,145),(19,146),(20,147),(21,148),(22,149),(23,150),(24,151),(25,152),(26,153),(27,154),(28,155),(29,156),(30,157),(31,158),(32,159),(33,160),(34,161),(35,162),(36,163),(37,164),(38,165),(39,166),(40,167),(41,168),(42,169),(43,170),(44,171),(45,172),(46,173),(47,174),(48,175),(49,176),(50,177),(51,178),(52,179),(53,180),(54,181),(55,182),(56,183),(57,184),(58,185),(59,186),(60,187),(61,188),(62,189),(63,190),(64,191),(65,192),(66,193),(67,194),(68,195),(69,196),(70,197),(71,198),(72,199),(73,200),(74,201),(75,202),(76,203),(77,204),(78,205),(79,206),(80,207),(81,208),(82,105),(83,106),(84,107),(85,108),(86,109),(87,110),(88,111),(89,112),(90,113),(91,114),(92,115),(93,116),(94,117),(95,118),(96,119),(97,120),(98,121),(99,122),(100,123),(101,124),(102,125),(103,126),(104,127)], [(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)], [(1,127),(2,126),(3,125),(4,124),(5,123),(6,122),(7,121),(8,120),(9,119),(10,118),(11,117),(12,116),(13,115),(14,114),(15,113),(16,112),(17,111),(18,110),(19,109),(20,108),(21,107),(22,106),(23,105),(24,208),(25,207),(26,206),(27,205),(28,204),(29,203),(30,202),(31,201),(32,200),(33,199),(34,198),(35,197),(36,196),(37,195),(38,194),(39,193),(40,192),(41,191),(42,190),(43,189),(44,188),(45,187),(46,186),(47,185),(48,184),(49,183),(50,182),(51,181),(52,180),(53,179),(54,178),(55,177),(56,176),(57,175),(58,174),(59,173),(60,172),(61,171),(62,170),(63,169),(64,168),(65,167),(66,166),(67,165),(68,164),(69,163),(70,162),(71,161),(72,160),(73,159),(74,158),(75,157),(76,156),(77,155),(78,154),(79,153),(80,152),(81,151),(82,150),(83,149),(84,148),(85,147),(86,146),(87,145),(88,144),(89,143),(90,142),(91,141),(92,140),(93,139),(94,138),(95,137),(96,136),(97,135),(98,134),(99,133),(100,132),(101,131),(102,130),(103,129),(104,128)])
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