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## G = C5⋊C8⋊D4order 320 = 26·5

### 1st semidirect product of C5⋊C8 and D4 acting via D4/C22=C2

Series: Derived Chief Lower central Upper central

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

Generators and relations for C5⋊C8⋊D4
G = < a,b,c,d | a5=b8=c4=d2=1, bab-1=a3, cac-1=a-1, ad=da, cbc-1=b5, bd=db, dcd=c-1 >

Subgroups: 442 in 124 conjugacy classes, 48 normal (42 characteristic)
C1, C2 [×3], C2 [×3], C4 [×6], C22, C22 [×2], C22 [×5], C5, C8 [×5], C2×C4 [×2], C2×C4 [×7], D4 [×2], C23, C23, D5, C10 [×3], C10 [×2], C42, C22⋊C4, C22⋊C4, C4⋊C4, C2×C8 [×6], M4(2) [×2], C22×C4 [×2], C2×D4, Dic5 [×2], Dic5 [×2], C20 [×2], D10 [×3], C2×C10, C2×C10 [×2], C2×C10 [×2], C8⋊C4, C22⋊C8 [×2], C4⋊C8, C4×D4, C22×C8, C2×M4(2), C5⋊C8 [×2], C5⋊C8 [×3], C4×D5 [×2], C2×Dic5 [×3], C2×Dic5 [×2], C5⋊D4 [×2], C2×C20 [×2], C22×D5, C22×C10, C89D4, C4×Dic5, C10.D4, D10⋊C4, C5×C22⋊C4, C4.F5 [×2], C2×C5⋊C8 [×4], C2×C5⋊C8 [×2], C2×C4×D5, C22×Dic5, C2×C5⋊D4, C20⋊C8, C10.C42, D10⋊C8, C23.2F5, Dic54D4, C2×C4.F5, C22×C5⋊C8, C5⋊C8⋊D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], C23, M4(2) [×2], C22×C4, C2×D4, C4○D4, F5, C4×D4, C2×M4(2), C8○D4, C2×F5 [×3], C89D4, C4.F5 [×2], C22×F5, C2×C4.F5, D4.F5, D4×F5, C5⋊C8⋊D4

Smallest permutation representation of C5⋊C8⋊D4
On 160 points
Generators in S160
```(1 65 82 36 159)(2 37 66 160 83)(3 153 38 84 67)(4 85 154 68 39)(5 69 86 40 155)(6 33 70 156 87)(7 157 34 88 71)(8 81 158 72 35)(9 55 136 44 102)(10 45 56 103 129)(11 104 46 130 49)(12 131 97 50 47)(13 51 132 48 98)(14 41 52 99 133)(15 100 42 134 53)(16 135 101 54 43)(17 127 114 93 63)(18 94 128 64 115)(19 57 95 116 121)(20 117 58 122 96)(21 123 118 89 59)(22 90 124 60 119)(23 61 91 120 125)(24 113 62 126 92)(25 139 74 150 112)(26 151 140 105 75)(27 106 152 76 141)(28 77 107 142 145)(29 143 78 146 108)(30 147 144 109 79)(31 110 148 80 137)(32 73 111 138 149)
(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 78 13 18)(2 75 14 23)(3 80 15 20)(4 77 16 17)(5 74 9 22)(6 79 10 19)(7 76 11 24)(8 73 12 21)(25 136 60 86)(26 133 61 83)(27 130 62 88)(28 135 63 85)(29 132 64 82)(30 129 57 87)(31 134 58 84)(32 131 59 81)(33 109 45 121)(34 106 46 126)(35 111 47 123)(36 108 48 128)(37 105 41 125)(38 110 42 122)(39 107 43 127)(40 112 44 124)(49 113 71 141)(50 118 72 138)(51 115 65 143)(52 120 66 140)(53 117 67 137)(54 114 68 142)(55 119 69 139)(56 116 70 144)(89 158 149 97)(90 155 150 102)(91 160 151 99)(92 157 152 104)(93 154 145 101)(94 159 146 98)(95 156 147 103)(96 153 148 100)
(1 18)(2 19)(3 20)(4 21)(5 22)(6 23)(7 24)(8 17)(9 74)(10 75)(11 76)(12 77)(13 78)(14 79)(15 80)(16 73)(25 44)(26 45)(27 46)(28 47)(29 48)(30 41)(31 42)(32 43)(33 61)(34 62)(35 63)(36 64)(37 57)(38 58)(39 59)(40 60)(49 152)(50 145)(51 146)(52 147)(53 148)(54 149)(55 150)(56 151)(65 94)(66 95)(67 96)(68 89)(69 90)(70 91)(71 92)(72 93)(81 127)(82 128)(83 121)(84 122)(85 123)(86 124)(87 125)(88 126)(97 142)(98 143)(99 144)(100 137)(101 138)(102 139)(103 140)(104 141)(105 129)(106 130)(107 131)(108 132)(109 133)(110 134)(111 135)(112 136)(113 157)(114 158)(115 159)(116 160)(117 153)(118 154)(119 155)(120 156)```

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

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

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

38 conjugacy classes

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

38 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 4 4 4 4 8 8 type + + + + + + + + + + + + - + image C1 C2 C2 C2 C2 C2 C2 C2 C4 C4 C4 C4 D4 C4○D4 M4(2) C8○D4 F5 C2×F5 C2×F5 C4.F5 D4.F5 D4×F5 kernel C5⋊C8⋊D4 C20⋊C8 C10.C42 D10⋊C8 C23.2F5 Dic5⋊4D4 C2×C4.F5 C22×C5⋊C8 C10.D4 D10⋊C4 C5×C22⋊C4 C2×C5⋊D4 C5⋊C8 Dic5 C2×C10 C10 C22⋊C4 C2×C4 C23 C22 C2 C2 # reps 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 1 2 1 4 1 1

Matrix representation of C5⋊C8⋊D4 in GL8(𝔽41)

 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 40 0 0 0 0 1 0 0 40 0 0 0 0 0 1 0 40 0 0 0 0 0 0 1 40
,
 38 2 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 9 32 0 0 0 0 0 0 9 32 0 9 0 0 0 0 9 0 32 9 0 0 0 0 0 0 32 9
,
 1 0 0 0 0 0 0 0 3 40 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 40 0 0 0 0 0 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 0 0 0
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40

`G:=sub<GL(8,GF(41))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,40,40,40,40],[38,0,0,0,0,0,0,0,2,3,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,9,9,9,0,0,0,0,0,32,32,0,0,0,0,0,0,0,0,32,32,0,0,0,0,0,9,9,9],[1,3,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40] >;`

C5⋊C8⋊D4 in GAP, Magma, Sage, TeX

`C_5\rtimes C_8\rtimes D_4`
`% in TeX`

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

`G:=SmallGroup(320,1031);`
`// by ID`

`G=gap.SmallGroup(320,1031);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,758,219,184,136,6278,1595]);`
`// 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,c*b*c^-1=b^5,b*d=d*b,d*c*d=c^-1>;`
`// generators/relations`

׿
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