metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D104⋊5C2, Q16⋊3D13, D26.7D4, C8.10D26, Q8.5D26, C104.8C22, C52.10C23, D52.5C22, Dic13.26D4, (C8×D13)⋊3C2, C13⋊4(C4○D8), Q8⋊D13⋊4C2, (C13×Q16)⋊3C2, C2.24(D4×D13), C26.36(C2×D4), D52⋊C2⋊3C2, C13⋊2C8.8C22, C4.10(C22×D13), (Q8×C13).5C22, (C4×D13).20C22, SmallGroup(416,140)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D104⋊C2
G = < a,b,c | a104=b2=c2=1, bab=a-1, cac=a25, cbc=a76b >
Subgroups: 512 in 62 conjugacy classes, 27 normal (17 characteristic)
C1, C2, C2, C4, C4, C22, C8, C8, C2×C4, D4, Q8, C13, C2×C8, D8, SD16, Q16, C4○D4, D13, C26, C4○D8, Dic13, C52, C52, D26, D26, C13⋊2C8, C104, C4×D13, C4×D13, D52, D52, Q8×C13, C8×D13, D104, Q8⋊D13, C13×Q16, D52⋊C2, D104⋊C2
Quotients: C1, C2, C22, D4, C23, C2×D4, D13, C4○D8, D26, C22×D13, D4×D13, D104⋊C2
►On 208 points
Generators in S208
(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)
(1 118)(2 117)(3 116)(4 115)(5 114)(6 113)(7 112)(8 111)(9 110)(10 109)(11 108)(12 107)(13 106)(14 105)(15 208)(16 207)(17 206)(18 205)(19 204)(20 203)(21 202)(22 201)(23 200)(24 199)(25 198)(26 197)(27 196)(28 195)(29 194)(30 193)(31 192)(32 191)(33 190)(34 189)(35 188)(36 187)(37 186)(38 185)(39 184)(40 183)(41 182)(42 181)(43 180)(44 179)(45 178)(46 177)(47 176)(48 175)(49 174)(50 173)(51 172)(52 171)(53 170)(54 169)(55 168)(56 167)(57 166)(58 165)(59 164)(60 163)(61 162)(62 161)(63 160)(64 159)(65 158)(66 157)(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)
(2 26)(3 51)(4 76)(5 101)(6 22)(7 47)(8 72)(9 97)(10 18)(11 43)(12 68)(13 93)(15 39)(16 64)(17 89)(19 35)(20 60)(21 85)(23 31)(24 56)(25 81)(28 52)(29 77)(30 102)(32 48)(33 73)(34 98)(36 44)(37 69)(38 94)(41 65)(42 90)(45 61)(46 86)(49 57)(50 82)(54 78)(55 103)(58 74)(59 99)(62 70)(63 95)(67 91)(71 87)(75 83)(80 104)(84 100)(88 96)(105 133)(106 158)(107 183)(108 208)(109 129)(110 154)(111 179)(112 204)(113 125)(114 150)(115 175)(116 200)(117 121)(118 146)(119 171)(120 196)(122 142)(123 167)(124 192)(126 138)(127 163)(128 188)(130 134)(131 159)(132 184)(135 155)(136 180)(137 205)(139 151)(140 176)(141 201)(143 147)(144 172)(145 197)(148 168)(149 193)(152 164)(153 189)(156 160)(157 185)(161 181)(162 206)(165 177)(166 202)(169 173)(170 198)(174 194)(178 190)(182 186)(187 207)(191 203)(195 199)
G:=sub<Sym(208)| (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), (1,118)(2,117)(3,116)(4,115)(5,114)(6,113)(7,112)(8,111)(9,110)(10,109)(11,108)(12,107)(13,106)(14,105)(15,208)(16,207)(17,206)(18,205)(19,204)(20,203)(21,202)(22,201)(23,200)(24,199)(25,198)(26,197)(27,196)(28,195)(29,194)(30,193)(31,192)(32,191)(33,190)(34,189)(35,188)(36,187)(37,186)(38,185)(39,184)(40,183)(41,182)(42,181)(43,180)(44,179)(45,178)(46,177)(47,176)(48,175)(49,174)(50,173)(51,172)(52,171)(53,170)(54,169)(55,168)(56,167)(57,166)(58,165)(59,164)(60,163)(61,162)(62,161)(63,160)(64,159)(65,158)(66,157)(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), (2,26)(3,51)(4,76)(5,101)(6,22)(7,47)(8,72)(9,97)(10,18)(11,43)(12,68)(13,93)(15,39)(16,64)(17,89)(19,35)(20,60)(21,85)(23,31)(24,56)(25,81)(28,52)(29,77)(30,102)(32,48)(33,73)(34,98)(36,44)(37,69)(38,94)(41,65)(42,90)(45,61)(46,86)(49,57)(50,82)(54,78)(55,103)(58,74)(59,99)(62,70)(63,95)(67,91)(71,87)(75,83)(80,104)(84,100)(88,96)(105,133)(106,158)(107,183)(108,208)(109,129)(110,154)(111,179)(112,204)(113,125)(114,150)(115,175)(116,200)(117,121)(118,146)(119,171)(120,196)(122,142)(123,167)(124,192)(126,138)(127,163)(128,188)(130,134)(131,159)(132,184)(135,155)(136,180)(137,205)(139,151)(140,176)(141,201)(143,147)(144,172)(145,197)(148,168)(149,193)(152,164)(153,189)(156,160)(157,185)(161,181)(162,206)(165,177)(166,202)(169,173)(170,198)(174,194)(178,190)(182,186)(187,207)(191,203)(195,199)>;
G:=Group( (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), (1,118)(2,117)(3,116)(4,115)(5,114)(6,113)(7,112)(8,111)(9,110)(10,109)(11,108)(12,107)(13,106)(14,105)(15,208)(16,207)(17,206)(18,205)(19,204)(20,203)(21,202)(22,201)(23,200)(24,199)(25,198)(26,197)(27,196)(28,195)(29,194)(30,193)(31,192)(32,191)(33,190)(34,189)(35,188)(36,187)(37,186)(38,185)(39,184)(40,183)(41,182)(42,181)(43,180)(44,179)(45,178)(46,177)(47,176)(48,175)(49,174)(50,173)(51,172)(52,171)(53,170)(54,169)(55,168)(56,167)(57,166)(58,165)(59,164)(60,163)(61,162)(62,161)(63,160)(64,159)(65,158)(66,157)(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), (2,26)(3,51)(4,76)(5,101)(6,22)(7,47)(8,72)(9,97)(10,18)(11,43)(12,68)(13,93)(15,39)(16,64)(17,89)(19,35)(20,60)(21,85)(23,31)(24,56)(25,81)(28,52)(29,77)(30,102)(32,48)(33,73)(34,98)(36,44)(37,69)(38,94)(41,65)(42,90)(45,61)(46,86)(49,57)(50,82)(54,78)(55,103)(58,74)(59,99)(62,70)(63,95)(67,91)(71,87)(75,83)(80,104)(84,100)(88,96)(105,133)(106,158)(107,183)(108,208)(109,129)(110,154)(111,179)(112,204)(113,125)(114,150)(115,175)(116,200)(117,121)(118,146)(119,171)(120,196)(122,142)(123,167)(124,192)(126,138)(127,163)(128,188)(130,134)(131,159)(132,184)(135,155)(136,180)(137,205)(139,151)(140,176)(141,201)(143,147)(144,172)(145,197)(148,168)(149,193)(152,164)(153,189)(156,160)(157,185)(161,181)(162,206)(165,177)(166,202)(169,173)(170,198)(174,194)(178,190)(182,186)(187,207)(191,203)(195,199) );
G=PermutationGroup([[(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)], [(1,118),(2,117),(3,116),(4,115),(5,114),(6,113),(7,112),(8,111),(9,110),(10,109),(11,108),(12,107),(13,106),(14,105),(15,208),(16,207),(17,206),(18,205),(19,204),(20,203),(21,202),(22,201),(23,200),(24,199),(25,198),(26,197),(27,196),(28,195),(29,194),(30,193),(31,192),(32,191),(33,190),(34,189),(35,188),(36,187),(37,186),(38,185),(39,184),(40,183),(41,182),(42,181),(43,180),(44,179),(45,178),(46,177),(47,176),(48,175),(49,174),(50,173),(51,172),(52,171),(53,170),(54,169),(55,168),(56,167),(57,166),(58,165),(59,164),(60,163),(61,162),(62,161),(63,160),(64,159),(65,158),(66,157),(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)], [(2,26),(3,51),(4,76),(5,101),(6,22),(7,47),(8,72),(9,97),(10,18),(11,43),(12,68),(13,93),(15,39),(16,64),(17,89),(19,35),(20,60),(21,85),(23,31),(24,56),(25,81),(28,52),(29,77),(30,102),(32,48),(33,73),(34,98),(36,44),(37,69),(38,94),(41,65),(42,90),(45,61),(46,86),(49,57),(50,82),(54,78),(55,103),(58,74),(59,99),(62,70),(63,95),(67,91),(71,87),(75,83),(80,104),(84,100),(88,96),(105,133),(106,158),(107,183),(108,208),(109,129),(110,154),(111,179),(112,204),(113,125),(114,150),(115,175),(116,200),(117,121),(118,146),(119,171),(120,196),(122,142),(123,167),(124,192),(126,138),(127,163),(128,188),(130,134),(131,159),(132,184),(135,155),(136,180),(137,205),(139,151),(140,176),(141,201),(143,147),(144,172),(145,197),(148,168),(149,193),(152,164),(153,189),(156,160),(157,185),(161,181),(162,206),(165,177),(166,202),(169,173),(170,198),(174,194),(178,190),(182,186),(187,207),(191,203),(195,199)]])
56 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 8A | 8B | 8C | 8D | 13A | ··· | 13F | 26A | ··· | 26F | 52A | ··· | 52F | 52G | ··· | 52R | 104A | ··· | 104L |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 13 | ··· | 13 | 26 | ··· | 26 | 52 | ··· | 52 | 52 | ··· | 52 | 104 | ··· | 104 |
| size | 1 | 1 | 26 | 52 | 52 | 2 | 4 | 4 | 13 | 13 | 2 | 2 | 26 | 26 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D13 | C4○D8 | D26 | D26 | D4×D13 | D104⋊C2 |
| kernel | D104⋊C2 | C8×D13 | D104 | Q8⋊D13 | C13×Q16 | D52⋊C2 | Dic13 | D26 | Q16 | C13 | C8 | Q8 | C2 | C1 |
| # reps | 1 | 1 | 1 | 2 | 1 | 2 | 1 | 1 | 6 | 4 | 6 | 12 | 6 | 12 |
Matrix representation of D104⋊C2 ►in GL4(𝔽313) generated by
| 13 | 37 | 0 | 0 |
| 276 | 280 | 0 | 0 |
| 0 | 0 | 125 | 0 |
| 0 | 0 | 182 | 308 |
| 300 | 276 | 0 | 0 |
| 13 | 13 | 0 | 0 |
| 0 | 0 | 262 | 260 |
| 0 | 0 | 303 | 51 |
| 1 | 0 | 0 | 0 |
| 261 | 312 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 128 | 312 |
G:=sub<GL(4,GF(313))| [13,276,0,0,37,280,0,0,0,0,125,182,0,0,0,308],[300,13,0,0,276,13,0,0,0,0,262,303,0,0,260,51],[1,261,0,0,0,312,0,0,0,0,1,128,0,0,0,312] >;
D104⋊C2 in GAP, Magma, Sage, TeX
D_{104}\rtimes C_2 % in TeX
G:=Group("D104:C2"); // GroupNames label
G:=SmallGroup(416,140);
// by ID
G=gap.SmallGroup(416,140);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-13,217,103,362,116,86,297,159,69,13829]);
// Polycyclic
G:=Group<a,b,c|a^104=b^2=c^2=1,b*a*b=a^-1,c*a*c=a^25,c*b*c=a^76*b>;
// generators/relations