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## G = C20.10C42order 320 = 26·5

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — C20.10C42
 Chief series C1 — C5 — C10 — Dic5 — C4×D5 — C2×C4×D5 — C2×C4.F5 — C20.10C42
 Lower central C5 — C10 — C20 — C20.10C42
 Upper central C1 — C22 — C2×C4 — C2×C8

Generators and relations for C20.10C42
G = < a,b,c | a20=1, b4=c4=a10, bab-1=a3, ac=ca, cbc-1=a15b >

Subgroups: 322 in 90 conjugacy classes, 38 normal (24 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, C23, D5, C10, C2×C8, C2×C8, M4(2), C22×C4, Dic5, C20, D10, D10, C2×C10, C22×C8, C2×M4(2), C52C8, C40, C5⋊C8, C4×D5, C2×Dic5, C2×C20, C22×D5, C4.C42, C8×D5, C2×C52C8, C2×C40, C4.F5, C4.F5, C2×C5⋊C8, C2×C4×D5, D5×C2×C8, C2×C4.F5, C20.10C42
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C42, C22⋊C4, C4⋊C4, F5, C2.C42, C8.C4, C2×F5, C4.C42, C4×F5, C4⋊F5, C22⋊F5, C40.C4, D10.Q8, D10.3Q8, C20.10C42

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

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

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

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

44 conjugacy classes

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

44 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 4 4 4 4 4 4 4 type + + + + + - + + + image C1 C2 C2 C4 C4 C4 D4 D4 Q8 C8.C4 F5 C2×F5 C4×F5 C22⋊F5 C4⋊F5 C40.C4 D10.Q8 kernel C20.10C42 D5×C2×C8 C2×C4.F5 C2×C5⋊2C8 C2×C40 C4.F5 C4×D5 C2×Dic5 C22×D5 C10 C2×C8 C2×C4 C4 C4 C22 C2 C2 # reps 1 1 2 2 2 8 2 1 1 8 1 1 2 2 2 4 4

Matrix representation of C20.10C42 in GL6(𝔽41)

 32 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 0 40 0 0 1 0 0 40 0 0 0 1 0 40 0 0 0 0 1 40
,
 0 1 0 0 0 0 9 0 0 0 0 0 0 0 7 0 34 21 0 0 0 21 13 28 0 0 20 28 13 21 0 0 20 21 34 0
,
 14 0 0 0 0 0 0 38 0 0 0 0 0 0 32 0 0 0 0 0 0 32 0 0 0 0 0 0 32 0 0 0 0 0 0 32

`G:=sub<GL(6,GF(41))| [32,0,0,0,0,0,0,9,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,40,40,40,40],[0,9,0,0,0,0,1,0,0,0,0,0,0,0,7,0,20,20,0,0,0,21,28,21,0,0,34,13,13,34,0,0,21,28,21,0],[14,0,0,0,0,0,0,38,0,0,0,0,0,0,32,0,0,0,0,0,0,32,0,0,0,0,0,0,32,0,0,0,0,0,0,32] >;`

C20.10C42 in GAP, Magma, Sage, TeX

`C_{20}._{10}C_4^2`
`% in TeX`

`G:=Group("C20.10C4^2");`
`// GroupNames label`

`G:=SmallGroup(320,234);`
`// by ID`

`G=gap.SmallGroup(320,234);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,176,184,1123,136,6278,3156]);`
`// Polycyclic`

`G:=Group<a,b,c|a^20=1,b^4=c^4=a^10,b*a*b^-1=a^3,a*c=c*a,c*b*c^-1=a^15*b>;`
`// generators/relations`

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