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## G = C42.165D10order 320 = 26·5

### 165th non-split extension by C42 of D10 acting via D10/C5=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C42.165D10
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C2×C4×D5 — D10.12D4 — C42.165D10
 Lower central C5 — C2×C10 — C42.165D10
 Upper central C1 — C22 — C42⋊2C2

Generators and relations for C42.165D10
G = < a,b,c,d | a4=b4=1, c10=d2=a2b2, ab=ba, cac-1=ab2, dad-1=a-1, cbc-1=a2b-1, dbd-1=b-1, dcd-1=c9 >

Subgroups: 678 in 196 conjugacy classes, 91 normal (all characteristic)
C1, C2 [×3], C2 [×2], C4 [×13], C22, C22 [×6], C5, C2×C4 [×6], C2×C4 [×9], D4, Q8 [×3], C23, C23, D5, C10 [×3], C10, C42, C42 [×2], C22⋊C4 [×3], C22⋊C4 [×7], C4⋊C4 [×3], C4⋊C4 [×13], C22×C4 [×2], C2×D4, C2×Q8 [×3], Dic5 [×7], C20 [×6], D10 [×3], C2×C10, C2×C10 [×3], C22⋊Q8 [×4], C22.D4 [×2], C4.4D4, C42.C2 [×2], C422C2, C422C2 [×3], C4⋊Q8 [×2], Dic10 [×3], C4×D5, C2×Dic5 [×7], C2×Dic5, C5⋊D4, C2×C20 [×6], C22×D5, C22×C10, C22.57C24, C4×Dic5 [×2], C10.D4 [×7], C4⋊Dic5 [×6], D10⋊C4 [×5], C23.D5 [×2], C4×C20, C5×C22⋊C4 [×3], C5×C4⋊C4 [×3], C2×Dic10 [×3], C2×C4×D5, C22×Dic5, C2×C5⋊D4, C202Q8, C422D5, Dic5.14D4 [×2], C23.D10, D10.12D4, Dic5.5D4, C22.D20, C20⋊Q8, Dic5.Q8, C4.Dic10, D10⋊Q8, D102Q8, C4⋊C4⋊D5, C5×C422C2, C42.165D10
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], D5, C24, D10 [×7], 2+ 1+4, 2- 1+4 [×2], C22×D5 [×7], C22.57C24, C23×D5, D48D10, D4.10D10 [×2], C42.165D10

Smallest permutation representation of C42.165D10
On 160 points
Generators in S160
```(1 73 36 94)(2 85 37 64)(3 75 38 96)(4 87 39 66)(5 77 40 98)(6 89 21 68)(7 79 22 100)(8 91 23 70)(9 61 24 82)(10 93 25 72)(11 63 26 84)(12 95 27 74)(13 65 28 86)(14 97 29 76)(15 67 30 88)(16 99 31 78)(17 69 32 90)(18 81 33 80)(19 71 34 92)(20 83 35 62)(41 160 104 140)(42 131 105 151)(43 142 106 122)(44 133 107 153)(45 144 108 124)(46 135 109 155)(47 146 110 126)(48 137 111 157)(49 148 112 128)(50 139 113 159)(51 150 114 130)(52 121 115 141)(53 152 116 132)(54 123 117 143)(55 154 118 134)(56 125 119 145)(57 156 120 136)(58 127 101 147)(59 158 102 138)(60 129 103 149)
(1 112 26 59)(2 103 27 50)(3 114 28 41)(4 105 29 52)(5 116 30 43)(6 107 31 54)(7 118 32 45)(8 109 33 56)(9 120 34 47)(10 111 35 58)(11 102 36 49)(12 113 37 60)(13 104 38 51)(14 115 39 42)(15 106 40 53)(16 117 21 44)(17 108 22 55)(18 119 23 46)(19 110 24 57)(20 101 25 48)(61 136 92 146)(62 127 93 157)(63 138 94 148)(64 129 95 159)(65 140 96 150)(66 131 97 141)(67 122 98 152)(68 133 99 143)(69 124 100 154)(70 135 81 145)(71 126 82 156)(72 137 83 147)(73 128 84 158)(74 139 85 149)(75 130 86 160)(76 121 87 151)(77 132 88 142)(78 123 89 153)(79 134 90 144)(80 125 91 155)
(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 10 11 20)(2 19 12 9)(3 8 13 18)(4 17 14 7)(5 6 15 16)(21 30 31 40)(22 39 32 29)(23 28 33 38)(24 37 34 27)(25 26 35 36)(41 109 51 119)(42 118 52 108)(43 107 53 117)(44 116 54 106)(45 105 55 115)(46 114 56 104)(47 103 57 113)(48 112 58 102)(49 101 59 111)(50 110 60 120)(61 64 71 74)(62 73 72 63)(65 80 75 70)(66 69 76 79)(67 78 77 68)(81 96 91 86)(82 85 92 95)(83 94 93 84)(87 90 97 100)(88 99 98 89)(121 144 131 154)(122 153 132 143)(123 142 133 152)(124 151 134 141)(125 160 135 150)(126 149 136 159)(127 158 137 148)(128 147 138 157)(129 156 139 146)(130 145 140 155)```

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

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

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

47 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A ··· 4F 4G ··· 4M 5A 5B 10A ··· 10F 10G 10H 20A ··· 20L 20M ··· 20R order 1 2 2 2 2 2 4 ··· 4 4 ··· 4 5 5 10 ··· 10 10 10 20 ··· 20 20 ··· 20 size 1 1 1 1 4 20 4 ··· 4 20 ··· 20 2 2 2 ··· 2 8 8 4 ··· 4 8 ··· 8

47 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + + + + + + + - + - image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 D5 D10 D10 D10 2+ 1+4 2- 1+4 D4⋊8D10 D4.10D10 kernel C42.165D10 C20⋊2Q8 C42⋊2D5 Dic5.14D4 C23.D10 D10.12D4 Dic5.5D4 C22.D20 C20⋊Q8 Dic5.Q8 C4.Dic10 D10⋊Q8 D10⋊2Q8 C4⋊C4⋊D5 C5×C42⋊2C2 C42⋊2C2 C42 C22⋊C4 C4⋊C4 C10 C10 C2 C2 # reps 1 1 1 2 1 1 1 1 1 1 1 1 1 1 1 2 2 6 6 1 2 4 8

Matrix representation of C42.165D10 in GL8(𝔽41)

 2 13 37 15 0 0 0 0 28 39 26 4 0 0 0 0 2 13 39 28 0 0 0 0 28 39 13 2 0 0 0 0 0 0 0 0 39 28 0 0 0 0 0 0 13 2 0 0 0 0 0 0 0 0 39 28 0 0 0 0 0 0 13 2
,
 1 0 39 0 0 0 0 0 0 1 0 39 0 0 0 0 1 0 40 0 0 0 0 0 0 1 0 40 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0
,
 3 3 39 39 0 0 0 0 38 21 2 27 0 0 0 0 2 2 38 38 0 0 0 0 39 14 3 20 0 0 0 0 0 0 0 0 13 13 0 0 0 0 0 0 28 9 0 0 0 0 0 0 0 0 28 28 0 0 0 0 0 0 13 32
,
 3 3 39 39 0 0 0 0 21 38 27 2 0 0 0 0 2 2 38 38 0 0 0 0 14 39 20 3 0 0 0 0 0 0 0 0 13 13 0 0 0 0 0 0 9 28 0 0 0 0 0 0 0 0 13 13 0 0 0 0 0 0 9 28

`G:=sub<GL(8,GF(41))| [2,28,2,28,0,0,0,0,13,39,13,39,0,0,0,0,37,26,39,13,0,0,0,0,15,4,28,2,0,0,0,0,0,0,0,0,39,13,0,0,0,0,0,0,28,2,0,0,0,0,0,0,0,0,39,13,0,0,0,0,0,0,28,2],[1,0,1,0,0,0,0,0,0,1,0,1,0,0,0,0,39,0,40,0,0,0,0,0,0,39,0,40,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[3,38,2,39,0,0,0,0,3,21,2,14,0,0,0,0,39,2,38,3,0,0,0,0,39,27,38,20,0,0,0,0,0,0,0,0,13,28,0,0,0,0,0,0,13,9,0,0,0,0,0,0,0,0,28,13,0,0,0,0,0,0,28,32],[3,21,2,14,0,0,0,0,3,38,2,39,0,0,0,0,39,27,38,20,0,0,0,0,39,2,38,3,0,0,0,0,0,0,0,0,13,9,0,0,0,0,0,0,13,28,0,0,0,0,0,0,0,0,13,9,0,0,0,0,0,0,13,28] >;`

C42.165D10 in GAP, Magma, Sage, TeX

`C_4^2._{165}D_{10}`
`% in TeX`

`G:=Group("C4^2.165D10");`
`// GroupNames label`

`G:=SmallGroup(320,1384);`
`// by ID`

`G=gap.SmallGroup(320,1384);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,758,219,1571,570,297,80,12550]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^4=b^4=1,c^10=d^2=a^2*b^2,a*b=b*a,c*a*c^-1=a*b^2,d*a*d^-1=a^-1,c*b*c^-1=a^2*b^-1,d*b*d^-1=b^-1,d*c*d^-1=c^9>;`
`// generators/relations`

׿
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