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## G = C20.30M4(2)  order 320 = 26·5

### 5th non-split extension by C20 of M4(2) acting via M4(2)/C22=C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C20.30M4(2)
 Chief series C1 — C5 — C10 — Dic5 — C2×Dic5 — C2×C5⋊C8 — C10.C42 — C20.30M4(2)
 Lower central C5 — C2×C10 — C20.30M4(2)
 Upper central C1 — C22 — C22×C4

Generators and relations for C20.30M4(2)
G = < a,b,c | a20=b8=1, c2=a10, bab-1=a3, ac=ca, cbc-1=b5 >

Subgroups: 330 in 110 conjugacy classes, 50 normal (30 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C5, C8, C2×C4, C2×C4, C23, C10, C10, C42, C2×C8, C22×C4, C22×C4, Dic5, Dic5, C20, C20, C2×C10, C2×C10, C2×C10, C8⋊C4, C22⋊C8, C4⋊C8, C2×C42, C5⋊C8, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×C10, C42.6C4, C4×Dic5, C2×C5⋊C8, C22×Dic5, C22×C20, C20⋊C8, C10.C42, C23.2F5, C2×C4×Dic5, C20.30M4(2)
Quotients: C1, C2, C4, C22, C2×C4, C23, M4(2), C22×C4, C4○D4, F5, C42⋊C2, C2×M4(2), C2×F5, C42.6C4, C4.F5, C22.F5, C22×F5, C2×C4.F5, D10.C23, C2×C22.F5, C20.30M4(2)

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

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

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

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

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 4 4 4 4 4 4 type + + + + + + + + - image C1 C2 C2 C2 C2 C4 C4 C4 C4○D4 M4(2) M4(2) F5 C2×F5 C2×F5 C22.F5 C4.F5 D10.C23 kernel C20.30M4(2) C20⋊C8 C10.C42 C23.2F5 C2×C4×Dic5 C4×Dic5 C22×Dic5 C22×C20 Dic5 C20 C2×C10 C22×C4 C2×C4 C23 C4 C22 C2 # reps 1 2 2 2 1 4 2 2 4 4 4 1 2 1 4 4 4

Matrix representation of C20.30M4(2) in GL6(𝔽41)

 1 39 0 0 0 0 1 40 0 0 0 0 0 0 0 32 0 0 0 0 9 19 0 0 0 0 16 6 9 19 0 0 11 5 22 19
,
 31 10 0 0 0 0 36 10 0 0 0 0 0 0 21 0 23 0 0 0 31 34 0 23 0 0 13 27 20 0 0 0 23 37 10 7
,
 40 2 0 0 0 0 40 1 0 0 0 0 0 0 9 0 0 0 0 0 0 9 0 0 0 0 0 0 9 0 0 0 0 0 0 9

`G:=sub<GL(6,GF(41))| [1,1,0,0,0,0,39,40,0,0,0,0,0,0,0,9,16,11,0,0,32,19,6,5,0,0,0,0,9,22,0,0,0,0,19,19],[31,36,0,0,0,0,10,10,0,0,0,0,0,0,21,31,13,23,0,0,0,34,27,37,0,0,23,0,20,10,0,0,0,23,0,7],[40,40,0,0,0,0,2,1,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,9] >;`

C20.30M4(2) in GAP, Magma, Sage, TeX

`C_{20}._{30}M_4(2)`
`% in TeX`

`G:=Group("C20.30M4(2)");`
`// GroupNames label`

`G:=SmallGroup(320,1097);`
`// by ID`

`G=gap.SmallGroup(320,1097);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,477,120,1094,136,6278,1595]);`
`// Polycyclic`

`G:=Group<a,b,c|a^20=b^8=1,c^2=a^10,b*a*b^-1=a^3,a*c=c*a,c*b*c^-1=b^5>;`
`// generators/relations`

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