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## G = C10.C42order 160 = 25·5

### 4th non-split extension by C10 of C42 acting via C42/C4=C4

Series: Derived Chief Lower central Upper central

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

Generators and relations for C10.C42
G = < a,b,c | a10=c4=1, b4=a5, bab-1=a3, ac=ca, cbc-1=a5b >

Character table of C10.C42

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 4F 4G 4H 5 8A 8B 8C 8D 8E 8F 8G 8H 10A 10B 10C 20A 20B 20C 20D size 1 1 1 1 2 2 5 5 5 5 10 10 4 10 10 10 10 10 10 10 10 4 4 4 4 4 4 4 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 -1 -1 1 1 1 1 -1 -1 1 -1 1 1 1 -1 -1 -1 1 1 1 1 -1 -1 -1 -1 linear of order 2 ρ3 1 1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 1 1 1 1 1 1 1 linear of order 2 ρ4 1 1 1 1 -1 -1 1 1 1 1 -1 -1 1 1 -1 -1 -1 1 1 1 -1 1 1 1 -1 -1 -1 -1 linear of order 2 ρ5 1 1 -1 -1 -i i 1 -1 1 -1 i -i 1 i 1 -1 1 -i i -i -1 1 -1 -1 i -i -i i linear of order 4 ρ6 1 1 -1 -1 -i i 1 -1 1 -1 i -i 1 -i -1 1 -1 i -i i 1 1 -1 -1 i -i -i i linear of order 4 ρ7 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 1 -i i -i -i i i -i i 1 1 1 1 1 1 1 linear of order 4 ρ8 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 1 i -i i i -i -i i -i 1 1 1 1 1 1 1 linear of order 4 ρ9 1 1 -1 -1 -i i -1 1 -1 1 -i i 1 1 i i -i 1 -1 -1 -i 1 -1 -1 i -i -i i linear of order 4 ρ10 1 1 -1 -1 i -i -1 1 -1 1 i -i 1 -1 i i -i -1 1 1 -i 1 -1 -1 -i i i -i linear of order 4 ρ11 1 1 -1 -1 i -i 1 -1 1 -1 -i i 1 -i 1 -1 1 i -i i -1 1 -1 -1 -i i i -i linear of order 4 ρ12 1 1 -1 -1 i -i 1 -1 1 -1 -i i 1 i -1 1 -1 -i i -i 1 1 -1 -1 -i i i -i linear of order 4 ρ13 1 1 1 1 -1 -1 -1 -1 -1 -1 1 1 1 -i -i i i i i -i -i 1 1 1 -1 -1 -1 -1 linear of order 4 ρ14 1 1 1 1 -1 -1 -1 -1 -1 -1 1 1 1 i i -i -i -i -i i i 1 1 1 -1 -1 -1 -1 linear of order 4 ρ15 1 1 -1 -1 i -i -1 1 -1 1 i -i 1 1 -i -i i 1 -1 -1 i 1 -1 -1 -i i i -i linear of order 4 ρ16 1 1 -1 -1 -i i -1 1 -1 1 -i i 1 -1 -i -i i -1 1 1 i 1 -1 -1 i -i -i i linear of order 4 ρ17 2 -2 2 -2 0 0 -2i -2i 2i 2i 0 0 2 0 0 0 0 0 0 0 0 -2 -2 2 0 0 0 0 complex lifted from M4(2) ρ18 2 -2 -2 2 0 0 2i -2i -2i 2i 0 0 2 0 0 0 0 0 0 0 0 -2 2 -2 0 0 0 0 complex lifted from M4(2) ρ19 2 -2 -2 2 0 0 -2i 2i 2i -2i 0 0 2 0 0 0 0 0 0 0 0 -2 2 -2 0 0 0 0 complex lifted from M4(2) ρ20 2 -2 2 -2 0 0 2i 2i -2i -2i 0 0 2 0 0 0 0 0 0 0 0 -2 -2 2 0 0 0 0 complex lifted from M4(2) ρ21 4 4 4 4 -4 -4 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 -1 -1 1 1 1 1 orthogonal lifted from C2×F5 ρ22 4 4 4 4 4 4 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from F5 ρ23 4 -4 4 -4 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 1 1 -1 √5 -√5 √5 -√5 symplectic lifted from C22.F5, Schur index 2 ρ24 4 -4 4 -4 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 1 1 -1 -√5 √5 -√5 √5 symplectic lifted from C22.F5, Schur index 2 ρ25 4 4 -4 -4 -4i 4i 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 1 1 -i i i -i complex lifted from C4×F5 ρ26 4 4 -4 -4 4i -4i 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 1 1 i -i -i i complex lifted from C4×F5 ρ27 4 -4 -4 4 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 1 -1 1 √-5 √-5 -√-5 -√-5 complex lifted from C4.F5 ρ28 4 -4 -4 4 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 1 -1 1 -√-5 -√-5 √-5 √-5 complex lifted from C4.F5

Smallest permutation representation of C10.C42
Regular action 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)
(1 106 22 84 6 101 27 89)(2 103 21 87 7 108 26 82)(3 110 30 90 8 105 25 85)(4 107 29 83 9 102 24 88)(5 104 28 86 10 109 23 81)(11 112 34 98 16 117 39 93)(12 119 33 91 17 114 38 96)(13 116 32 94 18 111 37 99)(14 113 31 97 19 118 36 92)(15 120 40 100 20 115 35 95)(41 142 64 128 46 147 69 123)(42 149 63 121 47 144 68 126)(43 146 62 124 48 141 67 129)(44 143 61 127 49 148 66 122)(45 150 70 130 50 145 65 125)(51 152 74 138 56 157 79 133)(52 159 73 131 57 154 78 136)(53 156 72 134 58 151 77 139)(54 153 71 137 59 158 76 132)(55 160 80 140 60 155 75 135)
(1 53 13 43)(2 54 14 44)(3 55 15 45)(4 56 16 46)(5 57 17 47)(6 58 18 48)(7 59 19 49)(8 60 20 50)(9 51 11 41)(10 52 12 42)(21 71 31 61)(22 72 32 62)(23 73 33 63)(24 74 34 64)(25 75 35 65)(26 76 36 66)(27 77 37 67)(28 78 38 68)(29 79 39 69)(30 80 40 70)(81 136 91 126)(82 137 92 127)(83 138 93 128)(84 139 94 129)(85 140 95 130)(86 131 96 121)(87 132 97 122)(88 133 98 123)(89 134 99 124)(90 135 100 125)(101 156 111 146)(102 157 112 147)(103 158 113 148)(104 159 114 149)(105 160 115 150)(106 151 116 141)(107 152 117 142)(108 153 118 143)(109 154 119 144)(110 155 120 145)```

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

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

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

Matrix representation of C10.C42 in GL6(𝔽41)

 40 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 1 40 0 0 0 0 1 0 0 0 40 0 1 0 0 0 0 40 1 0
,
 31 37 0 0 0 0 17 10 0 0 0 0 0 0 28 17 30 7 0 0 17 24 30 35 0 0 17 11 6 24 0 0 34 0 13 24
,
 30 4 0 0 0 0 31 11 0 0 0 0 0 0 19 3 0 38 0 0 0 22 3 38 0 0 38 3 22 0 0 0 38 0 3 19

`G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,1,1,1,1,0,0,40,0,0,0],[31,17,0,0,0,0,37,10,0,0,0,0,0,0,28,17,17,34,0,0,17,24,11,0,0,0,30,30,6,13,0,0,7,35,24,24],[30,31,0,0,0,0,4,11,0,0,0,0,0,0,19,0,38,38,0,0,3,22,3,0,0,0,0,3,22,3,0,0,38,38,0,19] >;`

C10.C42 in GAP, Magma, Sage, TeX

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

`G:=Group("C10.C4^2");`
`// GroupNames label`

`G:=SmallGroup(160,77);`
`// by ID`

`G=gap.SmallGroup(160,77);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-5,24,217,55,86,2309,1169]);`
`// Polycyclic`

`G:=Group<a,b,c|a^10=c^4=1,b^4=a^5,b*a*b^-1=a^3,a*c=c*a,c*b*c^-1=a^5*b>;`
`// generators/relations`

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