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## G = C20⋊M4(2)  order 320 = 26·5

### 1st semidirect product of C20 and M4(2) acting via M4(2)/C2=C2×C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C20⋊M4(2)
 Chief series C1 — C5 — C10 — Dic5 — C2×Dic5 — C2×C5⋊C8 — C4×C5⋊C8 — C20⋊M4(2)
 Lower central C5 — C2×C10 — C20⋊M4(2)
 Upper central C1 — C22 — C4⋊C4

Generators and relations for C20⋊M4(2)
G = < a,b,c | a20=b8=c2=1, bab-1=a7, cac=a-1, cbc=b5 >

Subgroups: 474 in 122 conjugacy classes, 48 normal (26 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×5], C22, C22 [×6], C5, C8 [×5], C2×C4, C2×C4 [×2], C2×C4 [×6], D4 [×2], C23 [×2], D5 [×2], C10 [×3], C42, C22⋊C4 [×2], C4⋊C4, C2×C8 [×4], M4(2) [×4], C22×C4 [×2], C2×D4, Dic5 [×2], Dic5, C20 [×2], C20 [×2], D10 [×6], C2×C10, C4×C8, C22⋊C8 [×2], C4⋊C8, C4×D4, C2×M4(2) [×2], C5⋊C8 [×2], C5⋊C8 [×3], C4×D5 [×4], D20 [×2], C2×Dic5 [×2], C2×C20, C2×C20 [×2], C22×D5 [×2], C86D4, C4×Dic5, D10⋊C4 [×2], C5×C4⋊C4, C4.F5 [×4], C2×C5⋊C8 [×2], C2×C5⋊C8 [×2], C2×C4×D5 [×2], C2×D20, C4×C5⋊C8, C20⋊C8, D10⋊C8 [×2], D208C4, C2×C4.F5 [×2], C20⋊M4(2)
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], C86D4, C4.F5 [×2], C22×F5, C2×C4.F5, D4×F5, Q8.F5, C20⋊M4(2)

Smallest permutation representation of C20⋊M4(2)
On 160 points
Generators in S160
```(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)
(1 139 57 28 109 69 144 83)(2 122 46 35 110 72 153 90)(3 125 55 22 111 75 142 97)(4 128 44 29 112 78 151 84)(5 131 53 36 113 61 160 91)(6 134 42 23 114 64 149 98)(7 137 51 30 115 67 158 85)(8 140 60 37 116 70 147 92)(9 123 49 24 117 73 156 99)(10 126 58 31 118 76 145 86)(11 129 47 38 119 79 154 93)(12 132 56 25 120 62 143 100)(13 135 45 32 101 65 152 87)(14 138 54 39 102 68 141 94)(15 121 43 26 103 71 150 81)(16 124 52 33 104 74 159 88)(17 127 41 40 105 77 148 95)(18 130 50 27 106 80 157 82)(19 133 59 34 107 63 146 89)(20 136 48 21 108 66 155 96)
(1 114)(2 113)(3 112)(4 111)(5 110)(6 109)(7 108)(8 107)(9 106)(10 105)(11 104)(12 103)(13 102)(14 101)(15 120)(16 119)(17 118)(18 117)(19 116)(20 115)(21 30)(22 29)(23 28)(24 27)(25 26)(31 40)(32 39)(33 38)(34 37)(35 36)(41 145)(42 144)(43 143)(44 142)(45 141)(46 160)(47 159)(48 158)(49 157)(50 156)(51 155)(52 154)(53 153)(54 152)(55 151)(56 150)(57 149)(58 148)(59 147)(60 146)(61 72)(62 71)(63 70)(64 69)(65 68)(66 67)(73 80)(74 79)(75 78)(76 77)(81 100)(82 99)(83 98)(84 97)(85 96)(86 95)(87 94)(88 93)(89 92)(90 91)(121 132)(122 131)(123 130)(124 129)(125 128)(126 127)(133 140)(134 139)(135 138)(136 137)```

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

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

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

38 conjugacy classes

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

38 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 4 4 4 8 8 type + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C4 C4 C4 D4 C4○D4 M4(2) C8○D4 F5 C2×F5 C4.F5 D4×F5 Q8.F5 kernel C20⋊M4(2) C4×C5⋊C8 C20⋊C8 D10⋊C8 D20⋊8C4 C2×C4.F5 D10⋊C4 C5×C4⋊C4 C2×D20 C5⋊C8 Dic5 C20 C10 C4⋊C4 C2×C4 C4 C2 C2 # reps 1 1 1 2 1 2 4 2 2 2 2 4 4 1 3 4 1 1

Matrix representation of C20⋊M4(2) in GL6(𝔽41)

 1 2 0 0 0 0 40 40 0 0 0 0 0 0 0 0 0 1 0 0 40 0 0 1 0 0 0 40 0 1 0 0 0 0 40 1
,
 40 39 0 0 0 0 0 1 0 0 0 0 0 0 21 8 2 2 0 0 23 16 23 10 0 0 31 18 25 18 0 0 39 39 33 20
,
 40 39 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0

`G:=sub<GL(6,GF(41))| [1,40,0,0,0,0,2,40,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,1,1,1,1],[40,0,0,0,0,0,39,1,0,0,0,0,0,0,21,23,31,39,0,0,8,16,18,39,0,0,2,23,25,33,0,0,2,10,18,20],[40,0,0,0,0,0,39,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1,0,0,0] >;`

C20⋊M4(2) in GAP, Magma, Sage, TeX

`C_{20}\rtimes M_4(2)`
`% in TeX`

`G:=Group("C20:M4(2)");`
`// GroupNames label`

`G:=SmallGroup(320,1043);`
`// by ID`

`G=gap.SmallGroup(320,1043);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,120,758,219,184,136,6278,1595]);`
`// Polycyclic`

`G:=Group<a,b,c|a^20=b^8=c^2=1,b*a*b^-1=a^7,c*a*c=a^-1,c*b*c=b^5>;`
`// generators/relations`

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