Copied to
clipboard

## G = C20.34M4(2)  order 320 = 26·5

### 9th non-split extension by C20 of M4(2) acting via M4(2)/C22=C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C20.34M4(2)
 Chief series C1 — C5 — C10 — Dic5 — C2×Dic5 — C2×C5⋊C8 — C23.2F5 — C20.34M4(2)
 Lower central C5 — C10 — C20.34M4(2)
 Upper central C1 — C2×C4 — C22×C4

Generators and relations for C20.34M4(2)
G = < a,b,c | a20=b8=c2=1, bab-1=a13, ac=ca, cbc=a10b5 >

Subgroups: 330 in 118 conjugacy classes, 58 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, C4×C8, C22⋊C8, C4⋊C8, C2×C42, C5⋊C8, C2×Dic5, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×C10, C42.12C4, C4×Dic5, C2×C5⋊C8, C22×Dic5, C22×C20, C4×C5⋊C8, Dic5⋊C8, C23.2F5, C2×C4×Dic5, C20.34M4(2)
Quotients: C1, C2, C4, C22, C8, C2×C4, C23, C2×C8, M4(2), C22×C4, C4○D4, F5, C42⋊C2, C22×C8, C2×M4(2), C2×F5, C42.12C4, D5⋊C8, C22.F5, C22×F5, C2×D5⋊C8, D10.C23, C2×C22.F5, C20.34M4(2)

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

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

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

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

56 conjugacy classes

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

56 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 4 4 4 4 4 4 type + + + + + + + + - image C1 C2 C2 C2 C2 C4 C4 C4 C8 C4○D4 M4(2) F5 C2×F5 C2×F5 C22.F5 D5⋊C8 D10.C23 kernel C20.34M4(2) C4×C5⋊C8 Dic5⋊C8 C23.2F5 C2×C4×Dic5 C4×Dic5 C22×Dic5 C22×C20 C2×Dic5 Dic5 C20 C22×C4 C2×C4 C23 C4 C22 C2 # reps 1 2 2 2 1 4 2 2 16 4 4 1 2 1 4 4 4

Matrix representation of C20.34M4(2) in GL6(𝔽41)

 32 0 0 0 0 0 0 32 0 0 0 0 0 0 7 40 0 0 0 0 8 40 0 0 0 0 32 7 0 35 0 0 33 14 7 34
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 3 21 39 0 0 0 27 20 0 39 0 0 34 29 38 20 0 0 2 20 14 21
,
 1 0 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 3 21 40 0 0 0 27 20 0 40

`G:=sub<GL(6,GF(41))| [32,0,0,0,0,0,0,32,0,0,0,0,0,0,7,8,32,33,0,0,40,40,7,14,0,0,0,0,0,7,0,0,0,0,35,34],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,3,27,34,2,0,0,21,20,29,20,0,0,39,0,38,14,0,0,0,39,20,21],[1,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,3,27,0,0,0,1,21,20,0,0,0,0,40,0,0,0,0,0,0,40] >;`

C20.34M4(2) in GAP, Magma, Sage, TeX

`C_{20}._{34}M_4(2)`
`% in TeX`

`G:=Group("C20.34M4(2)");`
`// GroupNames label`

`G:=SmallGroup(320,1092);`
`// by ID`

`G=gap.SmallGroup(320,1092);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,253,268,136,6278,1595]);`
`// Polycyclic`

`G:=Group<a,b,c|a^20=b^8=c^2=1,b*a*b^-1=a^13,a*c=c*a,c*b*c=a^10*b^5>;`
`// generators/relations`

׿
×
𝔽