direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C104⋊C2, C8⋊8D26, C4.6D52, C26⋊1SD16, C52.29D4, C104⋊9C22, C52.28C23, D52.6C22, C22.12D52, Dic26⋊3C22, (C2×C8)⋊5D13, (C2×C104)⋊7C2, C26.9(C2×D4), C13⋊1(C2×SD16), (C2×D52).5C2, C2.11(C2×D52), (C2×C26).16D4, (C2×C4).79D26, (C2×Dic26)⋊5C2, (C2×C52).88C22, C4.26(C22×D13), SmallGroup(416,123)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C104⋊C2
G = < a,b,c | a2=b104=c2=1, ab=ba, ac=ca, cbc=b51 >
Subgroups: 624 in 68 conjugacy classes, 33 normal (17 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, Q8, C23, C13, C2×C8, SD16, C2×D4, C2×Q8, D13, C26, C26, C2×SD16, Dic13, C52, D26, C2×C26, C104, Dic26, Dic26, D52, D52, C2×Dic13, C2×C52, C22×D13, C104⋊C2, C2×C104, C2×Dic26, C2×D52, C2×C104⋊C2
Quotients: C1, C2, C22, D4, C23, SD16, C2×D4, D13, C2×SD16, D26, D52, C22×D13, C104⋊C2, C2×D52, C2×C104⋊C2
►On 208 points
Generators in S208
(1 164)(2 165)(3 166)(4 167)(5 168)(6 169)(7 170)(8 171)(9 172)(10 173)(11 174)(12 175)(13 176)(14 177)(15 178)(16 179)(17 180)(18 181)(19 182)(20 183)(21 184)(22 185)(23 186)(24 187)(25 188)(26 189)(27 190)(28 191)(29 192)(30 193)(31 194)(32 195)(33 196)(34 197)(35 198)(36 199)(37 200)(38 201)(39 202)(40 203)(41 204)(42 205)(43 206)(44 207)(45 208)(46 105)(47 106)(48 107)(49 108)(50 109)(51 110)(52 111)(53 112)(54 113)(55 114)(56 115)(57 116)(58 117)(59 118)(60 119)(61 120)(62 121)(63 122)(64 123)(65 124)(66 125)(67 126)(68 127)(69 128)(70 129)(71 130)(72 131)(73 132)(74 133)(75 134)(76 135)(77 136)(78 137)(79 138)(80 139)(81 140)(82 141)(83 142)(84 143)(85 144)(86 145)(87 146)(88 147)(89 148)(90 149)(91 150)(92 151)(93 152)(94 153)(95 154)(96 155)(97 156)(98 157)(99 158)(100 159)(101 160)(102 161)(103 162)(104 163)
(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)
(1 164)(2 111)(3 162)(4 109)(5 160)(6 107)(7 158)(8 105)(9 156)(10 207)(11 154)(12 205)(13 152)(14 203)(15 150)(16 201)(17 148)(18 199)(19 146)(20 197)(21 144)(22 195)(23 142)(24 193)(25 140)(26 191)(27 138)(28 189)(29 136)(30 187)(31 134)(32 185)(33 132)(34 183)(35 130)(36 181)(37 128)(38 179)(39 126)(40 177)(41 124)(42 175)(43 122)(44 173)(45 120)(46 171)(47 118)(48 169)(49 116)(50 167)(51 114)(52 165)(53 112)(54 163)(55 110)(56 161)(57 108)(58 159)(59 106)(60 157)(61 208)(62 155)(63 206)(64 153)(65 204)(66 151)(67 202)(68 149)(69 200)(70 147)(71 198)(72 145)(73 196)(74 143)(75 194)(76 141)(77 192)(78 139)(79 190)(80 137)(81 188)(82 135)(83 186)(84 133)(85 184)(86 131)(87 182)(88 129)(89 180)(90 127)(91 178)(92 125)(93 176)(94 123)(95 174)(96 121)(97 172)(98 119)(99 170)(100 117)(101 168)(102 115)(103 166)(104 113)
G:=sub<Sym(208)| (1,164)(2,165)(3,166)(4,167)(5,168)(6,169)(7,170)(8,171)(9,172)(10,173)(11,174)(12,175)(13,176)(14,177)(15,178)(16,179)(17,180)(18,181)(19,182)(20,183)(21,184)(22,185)(23,186)(24,187)(25,188)(26,189)(27,190)(28,191)(29,192)(30,193)(31,194)(32,195)(33,196)(34,197)(35,198)(36,199)(37,200)(38,201)(39,202)(40,203)(41,204)(42,205)(43,206)(44,207)(45,208)(46,105)(47,106)(48,107)(49,108)(50,109)(51,110)(52,111)(53,112)(54,113)(55,114)(56,115)(57,116)(58,117)(59,118)(60,119)(61,120)(62,121)(63,122)(64,123)(65,124)(66,125)(67,126)(68,127)(69,128)(70,129)(71,130)(72,131)(73,132)(74,133)(75,134)(76,135)(77,136)(78,137)(79,138)(80,139)(81,140)(82,141)(83,142)(84,143)(85,144)(86,145)(87,146)(88,147)(89,148)(90,149)(91,150)(92,151)(93,152)(94,153)(95,154)(96,155)(97,156)(98,157)(99,158)(100,159)(101,160)(102,161)(103,162)(104,163), (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), (1,164)(2,111)(3,162)(4,109)(5,160)(6,107)(7,158)(8,105)(9,156)(10,207)(11,154)(12,205)(13,152)(14,203)(15,150)(16,201)(17,148)(18,199)(19,146)(20,197)(21,144)(22,195)(23,142)(24,193)(25,140)(26,191)(27,138)(28,189)(29,136)(30,187)(31,134)(32,185)(33,132)(34,183)(35,130)(36,181)(37,128)(38,179)(39,126)(40,177)(41,124)(42,175)(43,122)(44,173)(45,120)(46,171)(47,118)(48,169)(49,116)(50,167)(51,114)(52,165)(53,112)(54,163)(55,110)(56,161)(57,108)(58,159)(59,106)(60,157)(61,208)(62,155)(63,206)(64,153)(65,204)(66,151)(67,202)(68,149)(69,200)(70,147)(71,198)(72,145)(73,196)(74,143)(75,194)(76,141)(77,192)(78,139)(79,190)(80,137)(81,188)(82,135)(83,186)(84,133)(85,184)(86,131)(87,182)(88,129)(89,180)(90,127)(91,178)(92,125)(93,176)(94,123)(95,174)(96,121)(97,172)(98,119)(99,170)(100,117)(101,168)(102,115)(103,166)(104,113)>;
G:=Group( (1,164)(2,165)(3,166)(4,167)(5,168)(6,169)(7,170)(8,171)(9,172)(10,173)(11,174)(12,175)(13,176)(14,177)(15,178)(16,179)(17,180)(18,181)(19,182)(20,183)(21,184)(22,185)(23,186)(24,187)(25,188)(26,189)(27,190)(28,191)(29,192)(30,193)(31,194)(32,195)(33,196)(34,197)(35,198)(36,199)(37,200)(38,201)(39,202)(40,203)(41,204)(42,205)(43,206)(44,207)(45,208)(46,105)(47,106)(48,107)(49,108)(50,109)(51,110)(52,111)(53,112)(54,113)(55,114)(56,115)(57,116)(58,117)(59,118)(60,119)(61,120)(62,121)(63,122)(64,123)(65,124)(66,125)(67,126)(68,127)(69,128)(70,129)(71,130)(72,131)(73,132)(74,133)(75,134)(76,135)(77,136)(78,137)(79,138)(80,139)(81,140)(82,141)(83,142)(84,143)(85,144)(86,145)(87,146)(88,147)(89,148)(90,149)(91,150)(92,151)(93,152)(94,153)(95,154)(96,155)(97,156)(98,157)(99,158)(100,159)(101,160)(102,161)(103,162)(104,163), (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), (1,164)(2,111)(3,162)(4,109)(5,160)(6,107)(7,158)(8,105)(9,156)(10,207)(11,154)(12,205)(13,152)(14,203)(15,150)(16,201)(17,148)(18,199)(19,146)(20,197)(21,144)(22,195)(23,142)(24,193)(25,140)(26,191)(27,138)(28,189)(29,136)(30,187)(31,134)(32,185)(33,132)(34,183)(35,130)(36,181)(37,128)(38,179)(39,126)(40,177)(41,124)(42,175)(43,122)(44,173)(45,120)(46,171)(47,118)(48,169)(49,116)(50,167)(51,114)(52,165)(53,112)(54,163)(55,110)(56,161)(57,108)(58,159)(59,106)(60,157)(61,208)(62,155)(63,206)(64,153)(65,204)(66,151)(67,202)(68,149)(69,200)(70,147)(71,198)(72,145)(73,196)(74,143)(75,194)(76,141)(77,192)(78,139)(79,190)(80,137)(81,188)(82,135)(83,186)(84,133)(85,184)(86,131)(87,182)(88,129)(89,180)(90,127)(91,178)(92,125)(93,176)(94,123)(95,174)(96,121)(97,172)(98,119)(99,170)(100,117)(101,168)(102,115)(103,166)(104,113) );
G=PermutationGroup([[(1,164),(2,165),(3,166),(4,167),(5,168),(6,169),(7,170),(8,171),(9,172),(10,173),(11,174),(12,175),(13,176),(14,177),(15,178),(16,179),(17,180),(18,181),(19,182),(20,183),(21,184),(22,185),(23,186),(24,187),(25,188),(26,189),(27,190),(28,191),(29,192),(30,193),(31,194),(32,195),(33,196),(34,197),(35,198),(36,199),(37,200),(38,201),(39,202),(40,203),(41,204),(42,205),(43,206),(44,207),(45,208),(46,105),(47,106),(48,107),(49,108),(50,109),(51,110),(52,111),(53,112),(54,113),(55,114),(56,115),(57,116),(58,117),(59,118),(60,119),(61,120),(62,121),(63,122),(64,123),(65,124),(66,125),(67,126),(68,127),(69,128),(70,129),(71,130),(72,131),(73,132),(74,133),(75,134),(76,135),(77,136),(78,137),(79,138),(80,139),(81,140),(82,141),(83,142),(84,143),(85,144),(86,145),(87,146),(88,147),(89,148),(90,149),(91,150),(92,151),(93,152),(94,153),(95,154),(96,155),(97,156),(98,157),(99,158),(100,159),(101,160),(102,161),(103,162),(104,163)], [(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)], [(1,164),(2,111),(3,162),(4,109),(5,160),(6,107),(7,158),(8,105),(9,156),(10,207),(11,154),(12,205),(13,152),(14,203),(15,150),(16,201),(17,148),(18,199),(19,146),(20,197),(21,144),(22,195),(23,142),(24,193),(25,140),(26,191),(27,138),(28,189),(29,136),(30,187),(31,134),(32,185),(33,132),(34,183),(35,130),(36,181),(37,128),(38,179),(39,126),(40,177),(41,124),(42,175),(43,122),(44,173),(45,120),(46,171),(47,118),(48,169),(49,116),(50,167),(51,114),(52,165),(53,112),(54,163),(55,110),(56,161),(57,108),(58,159),(59,106),(60,157),(61,208),(62,155),(63,206),(64,153),(65,204),(66,151),(67,202),(68,149),(69,200),(70,147),(71,198),(72,145),(73,196),(74,143),(75,194),(76,141),(77,192),(78,139),(79,190),(80,137),(81,188),(82,135),(83,186),(84,133),(85,184),(86,131),(87,182),(88,129),(89,180),(90,127),(91,178),(92,125),(93,176),(94,123),(95,174),(96,121),(97,172),(98,119),(99,170),(100,117),(101,168),(102,115),(103,166),(104,113)]])
110 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 8A | 8B | 8C | 8D | 13A | ··· | 13F | 26A | ··· | 26R | 52A | ··· | 52X | 104A | ··· | 104AV |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 13 | ··· | 13 | 26 | ··· | 26 | 52 | ··· | 52 | 104 | ··· | 104 |
| size | 1 | 1 | 1 | 1 | 52 | 52 | 2 | 2 | 52 | 52 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
110 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | D4 | D4 | SD16 | D13 | D26 | D26 | D52 | D52 | C104⋊C2 |
| kernel | C2×C104⋊C2 | C104⋊C2 | C2×C104 | C2×Dic26 | C2×D52 | C52 | C2×C26 | C26 | C2×C8 | C8 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 1 | 1 | 4 | 6 | 12 | 6 | 12 | 12 | 48 |
Matrix representation of C2×C104⋊C2 ►in GL4(𝔽313) generated by
| 312 | 0 | 0 | 0 |
| 0 | 312 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 113 | 27 | 0 | 0 |
| 205 | 85 | 0 | 0 |
| 0 | 0 | 233 | 64 |
| 0 | 0 | 290 | 171 |
| 110 | 162 | 0 | 0 |
| 105 | 203 | 0 | 0 |
| 0 | 0 | 241 | 6 |
| 0 | 0 | 23 | 72 |
G:=sub<GL(4,GF(313))| [312,0,0,0,0,312,0,0,0,0,1,0,0,0,0,1],[113,205,0,0,27,85,0,0,0,0,233,290,0,0,64,171],[110,105,0,0,162,203,0,0,0,0,241,23,0,0,6,72] >;
C2×C104⋊C2 in GAP, Magma, Sage, TeX
C_2\times C_{104}\rtimes C_2 % in TeX
G:=Group("C2xC104:C2"); // GroupNames label
G:=SmallGroup(416,123);
// by ID
G=gap.SmallGroup(416,123);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-13,218,50,579,69,13829]);
// 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^51>;
// generators/relations