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## G = C20.34C42order 320 = 26·5

### 4th non-split extension by C20 of C42 acting via C42/C22=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — C20.34C42
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C22×C20 — C22×C5⋊2C8 — C20.34C42
 Lower central C5 — C10 — C20 — C20.34C42
 Upper central C1 — C2×C4 — C22×C4 — C2×M4(2)

Generators and relations for C20.34C42
G = < a,b,c | a20=1, b4=c4=a10, bab-1=a9, cac-1=a11, cbc-1=a15b >

Subgroups: 214 in 90 conjugacy classes, 51 normal (39 characteristic)
C1, C2, C2, C2, C4, C22, C22, C5, C8, C2×C4, C23, C10, C10, C10, C2×C8, M4(2), M4(2), C22×C4, C20, C2×C10, C2×C10, C22×C8, C2×M4(2), C2×M4(2), C52C8, C40, C2×C20, C22×C10, C4.C42, C2×C52C8, C2×C52C8, C4.Dic5, C4.Dic5, C2×C40, C5×M4(2), C5×M4(2), C22×C20, C22×C52C8, C2×C4.Dic5, C10×M4(2), C20.34C42
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, C20.53D4, C10.10C42, C20.34C42

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

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

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

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

68 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 5A 5B 8A 8B 8C 8D 8E ··· 8L 8M 8N 8O 8P 10A ··· 10F 10G 10H 10I 10J 20A ··· 20H 20I 20J 20K 20L 40A ··· 40P order 1 2 2 2 2 2 4 4 4 4 4 4 5 5 8 8 8 8 8 ··· 8 8 8 8 8 10 ··· 10 10 10 10 10 20 ··· 20 20 20 20 20 40 ··· 40 size 1 1 1 1 2 2 1 1 1 1 2 2 2 2 4 4 4 4 10 ··· 10 20 20 20 20 2 ··· 2 4 4 4 4 2 ··· 2 4 4 4 4 4 ··· 4

68 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 type + + + + + - + - + + - image C1 C2 C2 C2 C4 C4 C4 D4 Q8 D5 Dic5 D10 C8.C4 C4×D5 D20 C5⋊D4 Dic10 C20.53D4 kernel C20.34C42 C22×C5⋊2C8 C2×C4.Dic5 C10×M4(2) C2×C5⋊2C8 C4.Dic5 C5×M4(2) C2×C20 C22×C10 C2×M4(2) M4(2) C22×C4 C10 C2×C4 C2×C4 C2×C4 C23 C2 # reps 1 1 1 1 4 4 4 3 1 2 4 2 8 8 4 8 4 8

Matrix representation of C20.34C42 in GL4(𝔽41) generated by

 9 0 0 0 0 32 0 0 0 0 1 36 0 0 40 6
,
 3 0 0 0 0 14 0 0 0 0 2 27 0 0 9 39
,
 0 1 0 0 32 0 0 0 0 0 18 36 0 0 40 23
`G:=sub<GL(4,GF(41))| [9,0,0,0,0,32,0,0,0,0,1,40,0,0,36,6],[3,0,0,0,0,14,0,0,0,0,2,9,0,0,27,39],[0,32,0,0,1,0,0,0,0,0,18,40,0,0,36,23] >;`

C20.34C42 in GAP, Magma, Sage, TeX

`C_{20}._{34}C_4^2`
`% in TeX`

`G:=Group("C20.34C4^2");`
`// GroupNames label`

`G:=SmallGroup(320,116);`
`// by ID`

`G=gap.SmallGroup(320,116);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,365,36,184,570,136,12550]);`
`// Polycyclic`

`G:=Group<a,b,c|a^20=1,b^4=c^4=a^10,b*a*b^-1=a^9,c*a*c^-1=a^11,c*b*c^-1=a^15*b>;`
`// generators/relations`

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