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## G = C20⋊2M4(2)  order 320 = 26·5

### 2nd 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⋊2M4(2)
 Chief series C1 — C5 — C10 — Dic5 — C2×Dic5 — C2×C5⋊C8 — C2×C22.F5 — C20⋊2M4(2)
 Lower central C5 — C2×C10 — C20⋊2M4(2)
 Upper central C1 — C22 — C2×D4

Generators and relations for C202M4(2)
G = < a,b,c | a20=b8=c2=1, bab-1=a3, cac=a11, cbc=b5 >

Subgroups: 394 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 [×8], D4 [×2], C23 [×2], C10 [×3], C10 [×2], C42, C22⋊C4 [×2], C4⋊C4, C2×C8 [×4], M4(2) [×4], C22×C4 [×2], C2×D4, Dic5 [×2], Dic5 [×3], C20 [×2], C2×C10, C2×C10 [×6], C4×C8, C22⋊C8 [×2], C4⋊C8, C4×D4, C2×M4(2) [×2], C5⋊C8 [×2], C5⋊C8 [×3], C2×Dic5 [×2], C2×Dic5 [×2], C2×Dic5 [×4], C2×C20, C5×D4 [×2], C22×C10 [×2], C86D4, C4×Dic5, C4⋊Dic5, C23.D5 [×2], C2×C5⋊C8 [×2], C2×C5⋊C8 [×2], C22.F5 [×4], C22×Dic5 [×2], D4×C10, C4×C5⋊C8, C20⋊C8, C23.2F5 [×2], D4×Dic5, C2×C22.F5 [×2], C202M4(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, C22.F5 [×2], C22×F5, D4.F5, D4×F5, C2×C22.F5, C202M4(2)

Smallest permutation representation of C202M4(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)
(1 101 133 50 91 155 63 39)(2 108 122 53 92 142 72 22)(3 115 131 56 93 149 61 25)(4 102 140 59 94 156 70 28)(5 109 129 42 95 143 79 31)(6 116 138 45 96 150 68 34)(7 103 127 48 97 157 77 37)(8 110 136 51 98 144 66 40)(9 117 125 54 99 151 75 23)(10 104 134 57 100 158 64 26)(11 111 123 60 81 145 73 29)(12 118 132 43 82 152 62 32)(13 105 121 46 83 159 71 35)(14 112 130 49 84 146 80 38)(15 119 139 52 85 153 69 21)(16 106 128 55 86 160 78 24)(17 113 137 58 87 147 67 27)(18 120 126 41 88 154 76 30)(19 107 135 44 89 141 65 33)(20 114 124 47 90 148 74 36)
(1 6)(2 17)(3 8)(4 19)(5 10)(7 12)(9 14)(11 16)(13 18)(15 20)(21 47)(22 58)(23 49)(24 60)(25 51)(26 42)(27 53)(28 44)(29 55)(30 46)(31 57)(32 48)(33 59)(34 50)(35 41)(36 52)(37 43)(38 54)(39 45)(40 56)(61 66)(62 77)(63 68)(64 79)(65 70)(67 72)(69 74)(71 76)(73 78)(75 80)(81 86)(82 97)(83 88)(84 99)(85 90)(87 92)(89 94)(91 96)(93 98)(95 100)(101 150)(102 141)(103 152)(104 143)(105 154)(106 145)(107 156)(108 147)(109 158)(110 149)(111 160)(112 151)(113 142)(114 153)(115 144)(116 155)(117 146)(118 157)(119 148)(120 159)(121 126)(122 137)(123 128)(124 139)(125 130)(127 132)(129 134)(131 136)(133 138)(135 140)```

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

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

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

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 10D 10E 10F 10G 20A 20B 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 10 10 10 10 20 20 size 1 1 1 1 4 4 2 2 5 5 5 5 10 10 20 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 2 2 2 2 4 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 C2×F5 C22.F5 D4.F5 D4×F5 kernel C20⋊2M4(2) C4×C5⋊C8 C20⋊C8 C23.2F5 D4×Dic5 C2×C22.F5 C4⋊Dic5 C23.D5 D4×C10 C5⋊C8 Dic5 C20 C10 C2×D4 C2×C4 C23 C4 C2 C2 # reps 1 1 1 2 1 2 2 4 2 2 2 4 4 1 1 2 4 1 1

Matrix representation of C202M4(2) in GL8(𝔽41)

 1 39 0 0 0 0 0 0 1 40 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 1 0 0 0 0 40 0 0 1 0 0 0 0 0 40 0 1 0 0 0 0 0 0 40 1
,
 40 2 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 34 19 0 0 0 0 0 0 25 7 0 0 0 0 0 0 0 0 37 14 4 2 0 0 0 0 0 16 29 39 0 0 0 0 25 12 2 2 0 0 0 0 39 16 4 27
,
 40 2 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 18 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 0 0 0 0 0 0 0 0 40

`G:=sub<GL(8,GF(41))| [1,1,0,0,0,0,0,0,39,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,1,1,1,1],[40,0,0,0,0,0,0,0,2,1,0,0,0,0,0,0,0,0,34,25,0,0,0,0,0,0,19,7,0,0,0,0,0,0,0,0,37,0,25,39,0,0,0,0,14,16,12,16,0,0,0,0,4,29,2,4,0,0,0,0,2,39,2,27],[40,0,0,0,0,0,0,0,2,1,0,0,0,0,0,0,0,0,1,18,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40] >;`

C202M4(2) in GAP, Magma, Sage, TeX

`C_{20}\rtimes_2M_4(2)`
`% in TeX`

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

`G:=SmallGroup(320,1112);`
`// by ID`

`G=gap.SmallGroup(320,1112);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,477,758,219,136,6278,1595]);`
`// Polycyclic`

`G:=Group<a,b,c|a^20=b^8=c^2=1,b*a*b^-1=a^3,c*a*c=a^11,c*b*c=b^5>;`
`// generators/relations`

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