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## G = C5⋊C8⋊8D4order 320 = 26·5

### 1st semidirect product of C5⋊C8 and D4 acting through Inn(C5⋊C8)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C5⋊C8⋊8D4
 Chief series C1 — C5 — C10 — Dic5 — C2×Dic5 — C2×C5⋊C8 — C22×C5⋊C8 — C5⋊C8⋊8D4
 Lower central C5 — C10 — C5⋊C8⋊8D4
 Upper central C1 — C22 — C22⋊C4

Generators and relations for C5⋊C88D4
G = < a,b,c,d | a5=b8=c4=d2=1, bab-1=a3, cac-1=a-1, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 442 in 134 conjugacy classes, 56 normal (42 characteristic)
C1, C2 [×3], C2 [×4], C4 [×7], C22, C22 [×2], C22 [×6], C5, C8 [×5], C2×C4 [×2], C2×C4 [×7], D4 [×4], C23, C23, D5 [×2], C10 [×3], C10 [×2], C42, C22⋊C4, C22⋊C4, C4⋊C4, C2×C8 [×8], C22×C4 [×2], C2×D4, Dic5 [×2], Dic5 [×2], Dic5, C20 [×2], D10 [×2], D10 [×2], C2×C10, C2×C10 [×2], C2×C10 [×2], C4×C8, C22⋊C8 [×2], C4⋊C8, C4×D4, C22×C8 [×2], C5⋊C8 [×2], C5⋊C8 [×3], C4×D5 [×2], C2×Dic5 [×3], C2×Dic5 [×2], C5⋊D4 [×4], C2×C20 [×2], C22×D5, C22×C10, C8×D4, C4×Dic5, C10.D4, D10⋊C4, C5×C22⋊C4, D5⋊C8 [×2], C2×C5⋊C8 [×4], C2×C5⋊C8 [×2], C2×C4×D5, C22×Dic5, C2×C5⋊D4, C4×C5⋊C8, D10⋊C8, Dic5⋊C8, C23.2F5, Dic54D4, C2×D5⋊C8, C22×C5⋊C8, C5⋊C88D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C8 [×4], C2×C4 [×6], D4 [×2], C23, C2×C8 [×6], C22×C4, C2×D4, C4○D4, F5, C4×D4, C22×C8, C8○D4, C2×F5 [×3], C8×D4, D5⋊C8 [×2], C22×F5, C2×D5⋊C8, D4.F5, D4×F5, C5⋊C88D4

Smallest permutation representation of C5⋊C88D4
On 160 points
Generators in S160
```(1 12 52 130 113)(2 131 13 114 53)(3 115 132 54 14)(4 55 116 15 133)(5 16 56 134 117)(6 135 9 118 49)(7 119 136 50 10)(8 51 120 11 129)(17 106 93 87 66)(18 88 107 67 94)(19 68 81 95 108)(20 96 69 109 82)(21 110 89 83 70)(22 84 111 71 90)(23 72 85 91 112)(24 92 65 105 86)(25 57 48 125 102)(26 126 58 103 41)(27 104 127 42 59)(28 43 97 60 128)(29 61 44 121 98)(30 122 62 99 45)(31 100 123 46 63)(32 47 101 64 124)(33 73 143 157 147)(34 158 74 148 144)(35 149 159 137 75)(36 138 150 76 160)(37 77 139 153 151)(38 154 78 152 140)(39 145 155 141 79)(40 142 146 80 156)
(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)
(1 86 137 44)(2 87 138 45)(3 88 139 46)(4 81 140 47)(5 82 141 48)(6 83 142 41)(7 84 143 42)(8 85 144 43)(9 110 80 58)(10 111 73 59)(11 112 74 60)(12 105 75 61)(13 106 76 62)(14 107 77 63)(15 108 78 64)(16 109 79 57)(17 160 122 114)(18 153 123 115)(19 154 124 116)(20 155 125 117)(21 156 126 118)(22 157 127 119)(23 158 128 120)(24 159 121 113)(25 56 69 39)(26 49 70 40)(27 50 71 33)(28 51 72 34)(29 52 65 35)(30 53 66 36)(31 54 67 37)(32 55 68 38)(89 146 103 135)(90 147 104 136)(91 148 97 129)(92 149 98 130)(93 150 99 131)(94 151 100 132)(95 152 101 133)(96 145 102 134)
(1 44)(2 45)(3 46)(4 47)(5 48)(6 41)(7 42)(8 43)(9 126)(10 127)(11 128)(12 121)(13 122)(14 123)(15 124)(16 125)(17 76)(18 77)(19 78)(20 79)(21 80)(22 73)(23 74)(24 75)(25 134)(26 135)(27 136)(28 129)(29 130)(30 131)(31 132)(32 133)(33 90)(34 91)(35 92)(36 93)(37 94)(38 95)(39 96)(40 89)(49 103)(50 104)(51 97)(52 98)(53 99)(54 100)(55 101)(56 102)(57 117)(58 118)(59 119)(60 120)(61 113)(62 114)(63 115)(64 116)(65 149)(66 150)(67 151)(68 152)(69 145)(70 146)(71 147)(72 148)(81 140)(82 141)(83 142)(84 143)(85 144)(86 137)(87 138)(88 139)(105 159)(106 160)(107 153)(108 154)(109 155)(110 156)(111 157)(112 158)```

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

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

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

50 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 5 8A ··· 8H 8I ··· 8T 10A 10B 10C 10D 10E 20A 20B 20C 20D order 1 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 4 4 4 4 5 8 ··· 8 8 ··· 8 10 10 10 10 10 20 20 20 20 size 1 1 1 1 2 2 10 10 2 2 2 2 5 5 5 5 10 10 10 10 4 5 ··· 5 10 ··· 10 4 4 4 8 8 8 8 8 8

50 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 4 4 4 4 8 8 type + + + + + + + + + + + + - + image C1 C2 C2 C2 C2 C2 C2 C2 C4 C4 C4 C4 C8 D4 C4○D4 C8○D4 F5 C2×F5 C2×F5 D5⋊C8 D4.F5 D4×F5 kernel C5⋊C8⋊8D4 C4×C5⋊C8 D10⋊C8 Dic5⋊C8 C23.2F5 Dic5⋊4D4 C2×D5⋊C8 C22×C5⋊C8 C10.D4 D10⋊C4 C5×C22⋊C4 C2×C5⋊D4 C5⋊D4 C5⋊C8 Dic5 C10 C22⋊C4 C2×C4 C23 C22 C2 C2 # reps 1 1 1 1 1 1 1 1 2 2 2 2 16 2 2 4 1 2 1 4 1 1

Matrix representation of C5⋊C88D4 in GL6(𝔽41)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 40 0 0 1 0 0 40 0 0 0 1 0 40 0 0 0 0 1 40
,
 9 0 0 0 0 0 0 9 0 0 0 0 0 0 19 0 22 21 0 0 0 21 1 40 0 0 20 40 1 21 0 0 20 21 22 0
,
 0 1 0 0 0 0 40 0 0 0 0 0 0 0 40 1 0 0 0 0 0 1 0 0 0 0 0 1 0 40 0 0 0 1 40 0
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40

`G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,40,40,40,40],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,19,0,20,20,0,0,0,21,40,21,0,0,22,1,1,22,0,0,21,40,21,0],[0,40,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,0,1,1,1,1,0,0,0,0,0,40,0,0,0,0,40,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40] >;`

C5⋊C88D4 in GAP, Magma, Sage, TeX

`C_5\rtimes C_8\rtimes_8D_4`
`% in TeX`

`G:=Group("C5:C8:8D4");`
`// GroupNames label`

`G:=SmallGroup(320,1030);`
`// by ID`

`G=gap.SmallGroup(320,1030);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,219,184,136,6278,1595]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^5=b^8=c^4=d^2=1,b*a*b^-1=a^3,c*a*c^-1=a^-1,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;`
`// generators/relations`

׿
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