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## G = D104⋊C2order 416 = 25·13

### 5th semidirect product of D104 and C2 acting faithfully

Series: Derived Chief Lower central Upper central

 Derived series C1 — C52 — D104⋊C2
 Chief series C1 — C13 — C26 — C52 — C4×D13 — D52⋊C2 — D104⋊C2
 Lower central C13 — C26 — C52 — D104⋊C2
 Upper central C1 — C2 — C4 — Q16

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 [×3], C4, C4 [×3], C22 [×3], C8, C8, C2×C4 [×3], D4 [×4], Q8 [×2], C13, C2×C8, D8, SD16 [×2], Q16, C4○D4 [×2], D13 [×3], C26, C4○D8, Dic13, C52, C52 [×2], D26, D26 [×2], C132C8, C104, C4×D13, C4×D13 [×2], D52 [×2], D52 [×2], Q8×C13 [×2], C8×D13, D104, Q8⋊D13 [×2], C13×Q16, D52⋊C2 [×2], D104⋊C2
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, C2×D4, D13, C4○D8, D26 [×3], C22×D13, D4×D13, D104⋊C2

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

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

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

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

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`

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