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## G = D4⋊2D25order 400 = 24·52

### The semidirect product of D4 and D25 acting through Inn(D4)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C50 — D4⋊2D25
 Chief series C1 — C5 — C25 — C50 — D50 — C4×D25 — D4⋊2D25
 Lower central C25 — C50 — D4⋊2D25
 Upper central C1 — C2 — D4

Generators and relations for D42D25
G = < a,b,c,d | a4=b2=c25=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, dbd=a2b, dcd=c-1 >

Subgroups: 433 in 60 conjugacy classes, 29 normal (17 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C2×C4, D4, D4, Q8, D5, C10, C10, C4○D4, Dic5, C20, D10, C2×C10, C25, Dic10, C4×D5, C2×Dic5, C5⋊D4, C5×D4, D25, C50, C50, D42D5, Dic25, Dic25, C100, D50, C2×C50, Dic50, C4×D25, C2×Dic25, C25⋊D4, D4×C25, D42D25
Quotients: C1, C2, C22, C23, D5, C4○D4, D10, C22×D5, D25, D42D5, D50, C22×D25, D42D25

Smallest permutation representation of D42D25
On 200 points
Generators in S200
```(1 59 36 97)(2 60 37 98)(3 61 38 99)(4 62 39 100)(5 63 40 76)(6 64 41 77)(7 65 42 78)(8 66 43 79)(9 67 44 80)(10 68 45 81)(11 69 46 82)(12 70 47 83)(13 71 48 84)(14 72 49 85)(15 73 50 86)(16 74 26 87)(17 75 27 88)(18 51 28 89)(19 52 29 90)(20 53 30 91)(21 54 31 92)(22 55 32 93)(23 56 33 94)(24 57 34 95)(25 58 35 96)(101 151 136 178)(102 152 137 179)(103 153 138 180)(104 154 139 181)(105 155 140 182)(106 156 141 183)(107 157 142 184)(108 158 143 185)(109 159 144 186)(110 160 145 187)(111 161 146 188)(112 162 147 189)(113 163 148 190)(114 164 149 191)(115 165 150 192)(116 166 126 193)(117 167 127 194)(118 168 128 195)(119 169 129 196)(120 170 130 197)(121 171 131 198)(122 172 132 199)(123 173 133 200)(124 174 134 176)(125 175 135 177)
(51 89)(52 90)(53 91)(54 92)(55 93)(56 94)(57 95)(58 96)(59 97)(60 98)(61 99)(62 100)(63 76)(64 77)(65 78)(66 79)(67 80)(68 81)(69 82)(70 83)(71 84)(72 85)(73 86)(74 87)(75 88)(101 136)(102 137)(103 138)(104 139)(105 140)(106 141)(107 142)(108 143)(109 144)(110 145)(111 146)(112 147)(113 148)(114 149)(115 150)(116 126)(117 127)(118 128)(119 129)(120 130)(121 131)(122 132)(123 133)(124 134)(125 135)
(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)
(1 116)(2 115)(3 114)(4 113)(5 112)(6 111)(7 110)(8 109)(9 108)(10 107)(11 106)(12 105)(13 104)(14 103)(15 102)(16 101)(17 125)(18 124)(19 123)(20 122)(21 121)(22 120)(23 119)(24 118)(25 117)(26 136)(27 135)(28 134)(29 133)(30 132)(31 131)(32 130)(33 129)(34 128)(35 127)(36 126)(37 150)(38 149)(39 148)(40 147)(41 146)(42 145)(43 144)(44 143)(45 142)(46 141)(47 140)(48 139)(49 138)(50 137)(51 174)(52 173)(53 172)(54 171)(55 170)(56 169)(57 168)(58 167)(59 166)(60 165)(61 164)(62 163)(63 162)(64 161)(65 160)(66 159)(67 158)(68 157)(69 156)(70 155)(71 154)(72 153)(73 152)(74 151)(75 175)(76 189)(77 188)(78 187)(79 186)(80 185)(81 184)(82 183)(83 182)(84 181)(85 180)(86 179)(87 178)(88 177)(89 176)(90 200)(91 199)(92 198)(93 197)(94 196)(95 195)(96 194)(97 193)(98 192)(99 191)(100 190)```

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

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

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

70 conjugacy classes

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

70 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + - - image C1 C2 C2 C2 C2 C2 D5 C4○D4 D10 D10 D25 D50 D50 D4⋊2D5 D4⋊2D25 kernel D4⋊2D25 Dic50 C4×D25 C2×Dic25 C25⋊D4 D4×C25 C5×D4 C25 C20 C2×C10 D4 C4 C22 C5 C1 # reps 1 1 1 2 2 1 2 2 2 4 10 10 20 2 10

Matrix representation of D42D25 in GL4(𝔽101) generated by

 100 0 0 0 0 100 0 0 0 0 0 100 0 0 1 0
,
 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 100
,
 88 62 0 0 90 99 0 0 0 0 1 0 0 0 0 1
,
 56 20 0 0 20 45 0 0 0 0 0 91 0 0 10 0
`G:=sub<GL(4,GF(101))| [100,0,0,0,0,100,0,0,0,0,0,1,0,0,100,0],[1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,100],[88,90,0,0,62,99,0,0,0,0,1,0,0,0,0,1],[56,20,0,0,20,45,0,0,0,0,0,10,0,0,91,0] >;`

D42D25 in GAP, Magma, Sage, TeX

`D_4\rtimes_2D_{25}`
`% in TeX`

`G:=Group("D4:2D25");`
`// GroupNames label`

`G:=SmallGroup(400,40);`
`// by ID`

`G=gap.SmallGroup(400,40);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-5,-5,55,218,116,4324,628,11525]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^4=b^2=c^25=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=a^2*b,d*c*d=c^-1>;`
`// generators/relations`

׿
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