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## G = C20.40C42order 320 = 26·5

### 3rd non-split extension by C20 of C42 acting via C42/C2×C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — C20.40C42
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C22×C20 — C2×C4.Dic5 — C20.40C42
 Lower central C5 — C10 — C20 — C20.40C42
 Upper central C1 — C2×C4 — C22×C4 — C22×C8

Generators and relations for C20.40C42
G = < a,b,c | a20=1, b4=c4=a10, bab-1=a-1, ac=ca, cbc-1=a15b >

Subgroups: 214 in 90 conjugacy classes, 51 normal (21 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C5, C8, C2×C4, C2×C4, C23, C10, C10, C10, C2×C8, C2×C8, M4(2), C22×C4, C20, C20, C2×C10, C2×C10, C2×C10, C22×C8, C2×M4(2), C52C8, C40, C2×C20, C2×C20, C22×C10, C4.C42, C2×C52C8, C4.Dic5, C4.Dic5, C2×C40, C2×C40, C22×C20, C2×C4.Dic5, C22×C40, C20.40C42
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, D5, C42, C22⋊C4, C4⋊C4, Dic5, D10, C2.C42, C8.C4, Dic10, C4×D5, D20, C2×Dic5, C5⋊D4, C4.C42, C4×Dic5, C10.D4, C4⋊Dic5, D10⋊C4, C23.D5, C40.6C4, C10.10C42, C20.40C42

Smallest permutation representation of C20.40C42
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 94 16 99 11 84 6 89)(2 93 17 98 12 83 7 88)(3 92 18 97 13 82 8 87)(4 91 19 96 14 81 9 86)(5 90 20 95 15 100 10 85)(21 107 36 112 31 117 26 102)(22 106 37 111 32 116 27 101)(23 105 38 110 33 115 28 120)(24 104 39 109 34 114 29 119)(25 103 40 108 35 113 30 118)(41 144 56 149 51 154 46 159)(42 143 57 148 52 153 47 158)(43 142 58 147 53 152 48 157)(44 141 59 146 54 151 49 156)(45 160 60 145 55 150 50 155)(61 137 76 122 71 127 66 132)(62 136 77 121 72 126 67 131)(63 135 78 140 73 125 68 130)(64 134 79 139 74 124 69 129)(65 133 80 138 75 123 70 128)
(1 56 25 65 11 46 35 75)(2 57 26 66 12 47 36 76)(3 58 27 67 13 48 37 77)(4 59 28 68 14 49 38 78)(5 60 29 69 15 50 39 79)(6 41 30 70 16 51 40 80)(7 42 31 71 17 52 21 61)(8 43 32 72 18 53 22 62)(9 44 33 73 19 54 23 63)(10 45 34 74 20 55 24 64)(81 151 120 125 91 141 110 135)(82 152 101 126 92 142 111 136)(83 153 102 127 93 143 112 137)(84 154 103 128 94 144 113 138)(85 155 104 129 95 145 114 139)(86 156 105 130 96 146 115 140)(87 157 106 131 97 147 116 121)(88 158 107 132 98 148 117 122)(89 159 108 133 99 149 118 123)(90 160 109 134 100 150 119 124)```

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

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

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

92 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 5A 5B 8A ··· 8H 8I ··· 8P 10A ··· 10N 20A ··· 20P 40A ··· 40AF order 1 2 2 2 2 2 4 4 4 4 4 4 5 5 8 ··· 8 8 ··· 8 10 ··· 10 20 ··· 20 40 ··· 40 size 1 1 1 1 2 2 1 1 1 1 2 2 2 2 2 ··· 2 20 ··· 20 2 ··· 2 2 ··· 2 2 ··· 2

92 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 type + + + + - + - + + - image C1 C2 C2 C4 C4 D4 Q8 D5 Dic5 D10 C8.C4 C4×D5 D20 C5⋊D4 Dic10 C40.6C4 kernel C20.40C42 C2×C4.Dic5 C22×C40 C4.Dic5 C2×C40 C2×C20 C22×C10 C22×C8 C2×C8 C22×C4 C10 C2×C4 C2×C4 C2×C4 C23 C2 # reps 1 2 1 8 4 3 1 2 4 2 8 8 4 8 4 32

Matrix representation of C20.40C42 in GL3(𝔽41) generated by

 1 0 0 0 39 0 0 0 20
,
 32 0 0 0 0 1 0 32 0
,
 9 0 0 0 3 0 0 0 27
`G:=sub<GL(3,GF(41))| [1,0,0,0,39,0,0,0,20],[32,0,0,0,0,32,0,1,0],[9,0,0,0,3,0,0,0,27] >;`

C20.40C42 in GAP, Magma, Sage, TeX

`C_{20}._{40}C_4^2`
`% in TeX`

`G:=Group("C20.40C4^2");`
`// GroupNames label`

`G:=SmallGroup(320,110);`
`// by ID`

`G=gap.SmallGroup(320,110);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,176,184,1123,136,12550]);`
`// Polycyclic`

`G:=Group<a,b,c|a^20=1,b^4=c^4=a^10,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=a^15*b>;`
`// generators/relations`

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