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## G = C20⋊8M4(2)  order 320 = 26·5

### 1st semidirect product of C20 and M4(2) acting via M4(2)/C22=C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C20⋊8M4(2)
 Chief series C1 — C5 — C10 — Dic5 — C2×Dic5 — C2×C5⋊C8 — C20⋊C8 — C20⋊8M4(2)
 Lower central C5 — C2×C10 — C20⋊8M4(2)
 Upper central C1 — C22 — C22×C4

Generators and relations for C208M4(2)
G = < a,b,c | a20=b8=c2=1, bab-1=a3, ac=ca, cbc=b5 >

Subgroups: 378 in 126 conjugacy classes, 58 normal (32 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×8], C22, C22 [×2], C22 [×2], C5, C8 [×4], C2×C4 [×2], C2×C4 [×12], C23, C10 [×3], C10 [×2], C42 [×4], C2×C8 [×4], M4(2) [×4], C22×C4, C22×C4 [×2], Dic5 [×2], Dic5 [×4], Dic5, C20 [×2], C20, C2×C10, C2×C10 [×2], C2×C10 [×2], C4⋊C8 [×4], C2×C42, C2×M4(2) [×2], C5⋊C8 [×4], C2×Dic5 [×4], C2×Dic5 [×4], C2×Dic5 [×2], C2×C20 [×2], C2×C20 [×2], C22×C10, C4⋊M4(2), C4×Dic5 [×4], C2×C5⋊C8 [×4], C22.F5 [×4], C22×Dic5 [×2], C22×C20, C20⋊C8 [×2], Dic5⋊C8 [×2], C2×C4×Dic5, C2×C22.F5 [×2], C208M4(2)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], Q8 [×2], C23, C4⋊C4 [×4], M4(2) [×4], C22×C4, C2×D4, C2×Q8, F5, C2×C4⋊C4, C2×M4(2) [×2], C2×F5 [×3], C4⋊M4(2), C4⋊F5 [×2], C22.F5 [×2], C22×F5, D5⋊M4(2), C2×C4⋊F5, C2×C22.F5, C208M4(2)

Smallest permutation representation of C208M4(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)
(1 131 154 71 81 118 52 34)(2 138 143 74 82 105 41 37)(3 125 152 77 83 112 50 40)(4 132 141 80 84 119 59 23)(5 139 150 63 85 106 48 26)(6 126 159 66 86 113 57 29)(7 133 148 69 87 120 46 32)(8 140 157 72 88 107 55 35)(9 127 146 75 89 114 44 38)(10 134 155 78 90 101 53 21)(11 121 144 61 91 108 42 24)(12 128 153 64 92 115 51 27)(13 135 142 67 93 102 60 30)(14 122 151 70 94 109 49 33)(15 129 160 73 95 116 58 36)(16 136 149 76 96 103 47 39)(17 123 158 79 97 110 56 22)(18 130 147 62 98 117 45 25)(19 137 156 65 99 104 54 28)(20 124 145 68 100 111 43 31)
(21 78)(22 79)(23 80)(24 61)(25 62)(26 63)(27 64)(28 65)(29 66)(30 67)(31 68)(32 69)(33 70)(34 71)(35 72)(36 73)(37 74)(38 75)(39 76)(40 77)(101 134)(102 135)(103 136)(104 137)(105 138)(106 139)(107 140)(108 121)(109 122)(110 123)(111 124)(112 125)(113 126)(114 127)(115 128)(116 129)(117 130)(118 131)(119 132)(120 133)```

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

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

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

44 conjugacy classes

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

44 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 4 4 4 4 4 4 type + + + + + + - + + + - image C1 C2 C2 C2 C2 C4 C4 C4 D4 Q8 M4(2) M4(2) F5 C2×F5 C2×F5 C22.F5 C4⋊F5 D5⋊M4(2) kernel C20⋊8M4(2) C20⋊C8 Dic5⋊C8 C2×C4×Dic5 C2×C22.F5 C4×Dic5 C22×Dic5 C22×C20 C2×Dic5 C2×Dic5 Dic5 C20 C22×C4 C2×C4 C23 C4 C22 C2 # reps 1 2 2 1 2 4 2 2 2 2 4 4 1 2 1 4 4 4

Matrix representation of C208M4(2) in GL6(𝔽41)

 7 2 0 0 0 0 16 34 0 0 0 0 0 0 28 9 9 3 0 0 32 0 39 4 0 0 0 0 28 28 0 0 0 0 13 32
,
 0 35 0 0 0 0 7 0 0 0 0 0 0 0 38 14 31 17 0 0 37 33 28 0 0 0 13 8 0 28 0 0 4 28 3 11
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 29 20 0 0 0 1 40 36 0 0 0 0 40 0 0 0 0 0 0 40

`G:=sub<GL(6,GF(41))| [7,16,0,0,0,0,2,34,0,0,0,0,0,0,28,32,0,0,0,0,9,0,0,0,0,0,9,39,28,13,0,0,3,4,28,32],[0,7,0,0,0,0,35,0,0,0,0,0,0,0,38,37,13,4,0,0,14,33,8,28,0,0,31,28,0,3,0,0,17,0,28,11],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,29,40,40,0,0,0,20,36,0,40] >;`

C208M4(2) in GAP, Magma, Sage, TeX

`C_{20}\rtimes_8M_4(2)`
`% in TeX`

`G:=Group("C20:8M4(2)");`
`// GroupNames label`

`G:=SmallGroup(320,1096);`
`// by ID`

`G=gap.SmallGroup(320,1096);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,253,120,758,136,6278,1595]);`
`// Polycyclic`

`G:=Group<a,b,c|a^20=b^8=c^2=1,b*a*b^-1=a^3,a*c=c*a,c*b*c=b^5>;`
`// generators/relations`

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