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## G = Q8⋊D25order 400 = 24·52

### The semidirect product of Q8 and D25 acting via D25/C25=C2

Aliases: Q8⋊D25, C4.4D50, C253SD16, C50.10D4, C20.4D10, D100.2C2, C100.4C22, C252C83C2, C5.(Q8⋊D5), (Q8×C25)⋊1C2, (C5×Q8).2D5, C2.7(C25⋊D4), C10.17(C5⋊D4), SmallGroup(400,18)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C100 — Q8⋊D25
 Chief series C1 — C5 — C25 — C50 — C100 — D100 — Q8⋊D25
 Lower central C25 — C50 — C100 — Q8⋊D25
 Upper central C1 — C2 — C4 — Q8

Generators and relations for Q8⋊D25
G = < a,b,c,d | a4=c25=d2=1, b2=a2, bab-1=dad=a-1, ac=ca, bc=cb, dbd=a-1b, dcd=c-1 >

Smallest permutation representation of Q8⋊D25
On 200 points
Generators in S200
```(1 91 34 59)(2 92 35 60)(3 93 36 61)(4 94 37 62)(5 95 38 63)(6 96 39 64)(7 97 40 65)(8 98 41 66)(9 99 42 67)(10 100 43 68)(11 76 44 69)(12 77 45 70)(13 78 46 71)(14 79 47 72)(15 80 48 73)(16 81 49 74)(17 82 50 75)(18 83 26 51)(19 84 27 52)(20 85 28 53)(21 86 29 54)(22 87 30 55)(23 88 31 56)(24 89 32 57)(25 90 33 58)(101 171 136 194)(102 172 137 195)(103 173 138 196)(104 174 139 197)(105 175 140 198)(106 151 141 199)(107 152 142 200)(108 153 143 176)(109 154 144 177)(110 155 145 178)(111 156 146 179)(112 157 147 180)(113 158 148 181)(114 159 149 182)(115 160 150 183)(116 161 126 184)(117 162 127 185)(118 163 128 186)(119 164 129 187)(120 165 130 188)(121 166 131 189)(122 167 132 190)(123 168 133 191)(124 169 134 192)(125 170 135 193)
(1 149 34 114)(2 150 35 115)(3 126 36 116)(4 127 37 117)(5 128 38 118)(6 129 39 119)(7 130 40 120)(8 131 41 121)(9 132 42 122)(10 133 43 123)(11 134 44 124)(12 135 45 125)(13 136 46 101)(14 137 47 102)(15 138 48 103)(16 139 49 104)(17 140 50 105)(18 141 26 106)(19 142 27 107)(20 143 28 108)(21 144 29 109)(22 145 30 110)(23 146 31 111)(24 147 32 112)(25 148 33 113)(51 199 83 151)(52 200 84 152)(53 176 85 153)(54 177 86 154)(55 178 87 155)(56 179 88 156)(57 180 89 157)(58 181 90 158)(59 182 91 159)(60 183 92 160)(61 184 93 161)(62 185 94 162)(63 186 95 163)(64 187 96 164)(65 188 97 165)(66 189 98 166)(67 190 99 167)(68 191 100 168)(69 192 76 169)(70 193 77 170)(71 194 78 171)(72 195 79 172)(73 196 80 173)(74 197 81 174)(75 198 82 175)
(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 25)(2 24)(3 23)(4 22)(5 21)(6 20)(7 19)(8 18)(9 17)(10 16)(11 15)(12 14)(26 41)(27 40)(28 39)(29 38)(30 37)(31 36)(32 35)(33 34)(42 50)(43 49)(44 48)(45 47)(51 98)(52 97)(53 96)(54 95)(55 94)(56 93)(57 92)(58 91)(59 90)(60 89)(61 88)(62 87)(63 86)(64 85)(65 84)(66 83)(67 82)(68 81)(69 80)(70 79)(71 78)(72 77)(73 76)(74 100)(75 99)(101 171)(102 170)(103 169)(104 168)(105 167)(106 166)(107 165)(108 164)(109 163)(110 162)(111 161)(112 160)(113 159)(114 158)(115 157)(116 156)(117 155)(118 154)(119 153)(120 152)(121 151)(122 175)(123 174)(124 173)(125 172)(126 179)(127 178)(128 177)(129 176)(130 200)(131 199)(132 198)(133 197)(134 196)(135 195)(136 194)(137 193)(138 192)(139 191)(140 190)(141 189)(142 188)(143 187)(144 186)(145 185)(146 184)(147 183)(148 182)(149 181)(150 180)```

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

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

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

67 conjugacy classes

 class 1 2A 2B 4A 4B 5A 5B 8A 8B 10A 10B 20A ··· 20F 25A ··· 25J 50A ··· 50J 100A ··· 100AD order 1 2 2 4 4 5 5 8 8 10 10 20 ··· 20 25 ··· 25 50 ··· 50 100 ··· 100 size 1 1 100 2 4 2 2 50 50 2 2 4 ··· 4 2 ··· 2 2 ··· 2 4 ··· 4

67 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + image C1 C2 C2 C2 D4 D5 SD16 D10 C5⋊D4 D25 D50 C25⋊D4 Q8⋊D5 Q8⋊D25 kernel Q8⋊D25 C25⋊2C8 D100 Q8×C25 C50 C5×Q8 C25 C20 C10 Q8 C4 C2 C5 C1 # reps 1 1 1 1 1 2 2 2 4 10 10 20 2 10

Matrix representation of Q8⋊D25 in GL4(𝔽401) generated by

 1 0 0 0 0 1 0 0 0 0 400 160 0 0 5 1
,
 400 0 0 0 0 400 0 0 0 0 0 189 0 0 157 0
,
 162 373 0 0 28 62 0 0 0 0 1 0 0 0 0 1
,
 62 28 0 0 364 339 0 0 0 0 1 0 0 0 396 400
`G:=sub<GL(4,GF(401))| [1,0,0,0,0,1,0,0,0,0,400,5,0,0,160,1],[400,0,0,0,0,400,0,0,0,0,0,157,0,0,189,0],[162,28,0,0,373,62,0,0,0,0,1,0,0,0,0,1],[62,364,0,0,28,339,0,0,0,0,1,396,0,0,0,400] >;`

Q8⋊D25 in GAP, Magma, Sage, TeX

`Q_8\rtimes D_{25}`
`% in TeX`

`G:=Group("Q8:D25");`
`// GroupNames label`

`G:=SmallGroup(400,18);`
`// by ID`

`G=gap.SmallGroup(400,18);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-5,-5,73,55,218,116,50,4324,628,11525]);`
`// Polycyclic`

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

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