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## G = C10.10C42order 160 = 25·5

### 5th non-split extension by C10 of C42 acting via C42/C2×C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C10.10C42
 Chief series C1 — C5 — C10 — C2×C10 — C22×C10 — C22×Dic5 — C10.10C42
 Lower central C5 — C10 — C10.10C42
 Upper central C1 — C23 — C22×C4

Generators and relations for C10.10C42
G = < a,b,c | a10=b4=c4=1, bab-1=a-1, ac=ca, cbc-1=a5b >

Subgroups: 192 in 76 conjugacy classes, 45 normal (19 characteristic)
C1, C2, C2, C4, C22, C22, C5, C2×C4, C2×C4, C23, C10, C10, C22×C4, C22×C4, Dic5, C20, C2×C10, C2×C10, C2.C42, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×C10, C22×Dic5, C22×C20, C10.10C42
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, D5, C42, C22⋊C4, C4⋊C4, Dic5, D10, C2.C42, Dic10, C4×D5, D20, C2×Dic5, C5⋊D4, C4×Dic5, C10.D4, C4⋊Dic5, D10⋊C4, C23.D5, C10.10C42

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

52 conjugacy classes

 class 1 2A ··· 2G 4A 4B 4C 4D 4E ··· 4L 5A 5B 10A ··· 10N 20A ··· 20P order 1 2 ··· 2 4 4 4 4 4 ··· 4 5 5 10 ··· 10 20 ··· 20 size 1 1 ··· 1 2 2 2 2 10 ··· 10 2 2 2 ··· 2 2 ··· 2

52 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 type + + + + - + - + - + image C1 C2 C2 C4 C4 D4 Q8 D5 Dic5 D10 Dic10 C4×D5 D20 C5⋊D4 kernel C10.10C42 C22×Dic5 C22×C20 C2×Dic5 C2×C20 C2×C10 C2×C10 C22×C4 C2×C4 C23 C22 C22 C22 C22 # reps 1 2 1 8 4 3 1 2 4 2 4 8 4 8

Matrix representation of C10.10C42 in GL4(𝔽41) generated by

 1 0 0 0 0 1 0 0 0 0 0 34 0 0 6 35
,
 9 0 0 0 0 32 0 0 0 0 23 17 0 0 5 18
,
 9 0 0 0 0 40 0 0 0 0 39 32 0 0 37 2
`G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,0,6,0,0,34,35],[9,0,0,0,0,32,0,0,0,0,23,5,0,0,17,18],[9,0,0,0,0,40,0,0,0,0,39,37,0,0,32,2] >;`

C10.10C42 in GAP, Magma, Sage, TeX

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

`G:=Group("C10.10C4^2");`
`// GroupNames label`

`G:=SmallGroup(160,38);`
`// by ID`

`G=gap.SmallGroup(160,38);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-5,24,217,55,4613]);`
`// Polycyclic`

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

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