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## G = D100⋊5C2order 400 = 24·52

### The semidirect product of D100 and C2 acting through Inn(D100)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C50 — D100⋊5C2
 Chief series C1 — C5 — C25 — C50 — D50 — C4×D25 — D100⋊5C2
 Lower central C25 — C50 — D100⋊5C2
 Upper central C1 — C4 — C2×C4

Generators and relations for D1005C2
G = < a,b,c | a100=b2=c2=1, bab=a-1, ac=ca, cbc=a50b >

Subgroups: 489 in 60 conjugacy classes, 29 normal (21 characteristic)
C1, C2, C2 [×3], C4 [×2], C4 [×2], C22, C22 [×2], C5, C2×C4, C2×C4 [×2], D4 [×3], Q8, D5 [×2], C10, C10, C4○D4, Dic5 [×2], C20 [×2], D10 [×2], C2×C10, C25, Dic10, C4×D5 [×2], D20, C5⋊D4 [×2], C2×C20, D25 [×2], C50, C50, C4○D20, Dic25 [×2], C100 [×2], D50 [×2], C2×C50, Dic50, C4×D25 [×2], D100, C25⋊D4 [×2], C2×C100, D1005C2
Quotients: C1, C2 [×7], C22 [×7], C23, D5, C4○D4, D10 [×3], C22×D5, D25, C4○D20, D50 [×3], C22×D25, D1005C2

Smallest permutation representation of D1005C2
On 200 points
Generators in S200
```(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)
(1 100)(2 99)(3 98)(4 97)(5 96)(6 95)(7 94)(8 93)(9 92)(10 91)(11 90)(12 89)(13 88)(14 87)(15 86)(16 85)(17 84)(18 83)(19 82)(20 81)(21 80)(22 79)(23 78)(24 77)(25 76)(26 75)(27 74)(28 73)(29 72)(30 71)(31 70)(32 69)(33 68)(34 67)(35 66)(36 65)(37 64)(38 63)(39 62)(40 61)(41 60)(42 59)(43 58)(44 57)(45 56)(46 55)(47 54)(48 53)(49 52)(50 51)(101 148)(102 147)(103 146)(104 145)(105 144)(106 143)(107 142)(108 141)(109 140)(110 139)(111 138)(112 137)(113 136)(114 135)(115 134)(116 133)(117 132)(118 131)(119 130)(120 129)(121 128)(122 127)(123 126)(124 125)(149 200)(150 199)(151 198)(152 197)(153 196)(154 195)(155 194)(156 193)(157 192)(158 191)(159 190)(160 189)(161 188)(162 187)(163 186)(164 185)(165 184)(166 183)(167 182)(168 181)(169 180)(170 179)(171 178)(172 177)(173 176)(174 175)
(1 200)(2 101)(3 102)(4 103)(5 104)(6 105)(7 106)(8 107)(9 108)(10 109)(11 110)(12 111)(13 112)(14 113)(15 114)(16 115)(17 116)(18 117)(19 118)(20 119)(21 120)(22 121)(23 122)(24 123)(25 124)(26 125)(27 126)(28 127)(29 128)(30 129)(31 130)(32 131)(33 132)(34 133)(35 134)(36 135)(37 136)(38 137)(39 138)(40 139)(41 140)(42 141)(43 142)(44 143)(45 144)(46 145)(47 146)(48 147)(49 148)(50 149)(51 150)(52 151)(53 152)(54 153)(55 154)(56 155)(57 156)(58 157)(59 158)(60 159)(61 160)(62 161)(63 162)(64 163)(65 164)(66 165)(67 166)(68 167)(69 168)(70 169)(71 170)(72 171)(73 172)(74 173)(75 174)(76 175)(77 176)(78 177)(79 178)(80 179)(81 180)(82 181)(83 182)(84 183)(85 184)(86 185)(87 186)(88 187)(89 188)(90 189)(91 190)(92 191)(93 192)(94 193)(95 194)(96 195)(97 196)(98 197)(99 198)(100 199)```

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

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

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

106 conjugacy classes

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 5A 5B 10A ··· 10F 20A ··· 20H 25A ··· 25J 50A ··· 50AD 100A ··· 100AN order 1 2 2 2 2 4 4 4 4 4 5 5 10 ··· 10 20 ··· 20 25 ··· 25 50 ··· 50 100 ··· 100 size 1 1 2 50 50 1 1 2 50 50 2 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2

106 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 type + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 D5 C4○D4 D10 D10 D25 C4○D20 D50 D50 D100⋊5C2 kernel D100⋊5C2 Dic50 C4×D25 D100 C25⋊D4 C2×C100 C2×C20 C25 C20 C2×C10 C2×C4 C5 C4 C22 C1 # reps 1 1 2 1 2 1 2 2 4 2 10 8 20 10 40

Matrix representation of D1005C2 in GL4(𝔽101) generated by

 4 99 0 0 59 97 0 0 0 0 51 46 0 0 57 95
,
 4 99 0 0 58 97 0 0 0 0 68 97 0 0 70 33
,
 40 81 0 0 85 61 0 0 0 0 100 0 0 0 0 100
`G:=sub<GL(4,GF(101))| [4,59,0,0,99,97,0,0,0,0,51,57,0,0,46,95],[4,58,0,0,99,97,0,0,0,0,68,70,0,0,97,33],[40,85,0,0,81,61,0,0,0,0,100,0,0,0,0,100] >;`

D1005C2 in GAP, Magma, Sage, TeX

`D_{100}\rtimes_5C_2`
`% in TeX`

`G:=Group("D100:5C2");`
`// GroupNames label`

`G:=SmallGroup(400,38);`
`// by ID`

`G=gap.SmallGroup(400,38);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-5,-5,55,218,4324,628,11525]);`
`// Polycyclic`

`G:=Group<a,b,c|a^100=b^2=c^2=1,b*a*b=a^-1,a*c=c*a,c*b*c=a^50*b>;`
`// generators/relations`

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