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## G = C42.116D10order 320 = 26·5

### 116th non-split extension by C42 of D10 acting via D10/C5=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C42.116D10
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C2×C4×D5 — D10.12D4 — C42.116D10
 Lower central C5 — C2×C10 — C42.116D10
 Upper central C1 — C22 — C4×D4

Generators and relations for C42.116D10
G = < a,b,c,d | a4=b4=c10=1, d2=a2, ab=ba, cac-1=dad-1=a-1b2, bc=cb, dbd-1=a2b-1, dcd-1=a2c-1 >

Subgroups: 958 in 240 conjugacy classes, 95 normal (43 characteristic)
C1, C2 [×3], C2 [×5], C4 [×2], C4 [×9], C22, C22 [×15], C5, C2×C4 [×3], C2×C4 [×2], C2×C4 [×11], D4 [×12], C23 [×2], C23 [×3], D5 [×3], C10 [×3], C10 [×2], C42, C42, C22⋊C4 [×2], C22⋊C4 [×8], C4⋊C4, C4⋊C4 [×7], C22×C4 [×2], C22×C4 [×3], C2×D4, C2×D4 [×9], Dic5 [×5], C20 [×2], C20 [×4], D10 [×9], C2×C10, C2×C10 [×6], C42⋊C2, C4×D4, C4×D4, C4⋊D4 [×6], C22.D4 [×4], C42.C2, C41D4, C4×D5 [×4], D20 [×4], C2×Dic5 [×3], C2×Dic5 [×2], C5⋊D4 [×6], C2×C20 [×3], C2×C20 [×2], C2×C20 [×2], C5×D4 [×2], C22×D5, C22×D5 [×2], C22×C10 [×2], C22.34C24, C4×Dic5, C10.D4 [×2], C10.D4 [×2], C4⋊Dic5, C4⋊Dic5 [×2], D10⋊C4 [×2], D10⋊C4 [×4], C23.D5 [×2], C4×C20, C5×C22⋊C4 [×2], C5×C4⋊C4, C2×C4×D5, C2×C4×D5 [×2], C2×D20, C2×D20 [×2], C2×C5⋊D4 [×6], C22×C20 [×2], D4×C10, C42⋊D5, C4×D20, D10.12D4 [×2], D10⋊D4 [×2], C4.Dic10, C4⋊D20, C23.23D10 [×2], C207D4 [×2], C202D4, C20⋊D4, D4×C20, C42.116D10
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], D5, C4○D4 [×2], C24, D10 [×7], C2×C4○D4, 2+ 1+4 [×2], C22×D5 [×7], C22.34C24, C4○D20 [×2], C23×D5, C2×C4○D20, D46D10, D48D10, C42.116D10

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

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

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

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

62 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 4A ··· 4F 4G 4H 4I ··· 4M 5A 5B 10A ··· 10F 10G ··· 10N 20A ··· 20H 20I ··· 20X order 1 2 2 2 2 2 2 2 2 4 ··· 4 4 4 4 ··· 4 5 5 10 ··· 10 10 ··· 10 20 ··· 20 20 ··· 20 size 1 1 1 1 4 4 20 20 20 2 ··· 2 4 4 20 ··· 20 2 2 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4

62 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 D5 C4○D4 D10 D10 D10 D10 D10 C4○D20 2+ 1+4 D4⋊6D10 D4⋊8D10 kernel C42.116D10 C42⋊D5 C4×D20 D10.12D4 D10⋊D4 C4.Dic10 C4⋊D20 C23.23D10 C20⋊7D4 C20⋊2D4 C20⋊D4 D4×C20 C4×D4 C20 C42 C22⋊C4 C4⋊C4 C22×C4 C2×D4 C4 C10 C2 C2 # reps 1 1 1 2 2 1 1 2 2 1 1 1 2 4 2 4 2 4 2 16 2 4 4

Matrix representation of C42.116D10 in GL6(𝔽41)

 32 0 0 0 0 0 0 32 0 0 0 0 0 0 30 0 9 0 0 0 0 30 0 9 0 0 32 0 11 0 0 0 0 32 0 11
,
 32 0 0 0 0 0 0 32 0 0 0 0 0 0 24 40 0 0 0 0 1 17 0 0 0 0 0 0 24 40 0 0 0 0 1 17
,
 17 39 0 0 0 0 21 24 0 0 0 0 0 0 0 0 40 7 0 0 0 0 34 7 0 0 40 7 0 0 0 0 34 7 0 0
,
 17 39 0 0 0 0 22 24 0 0 0 0 0 0 14 11 33 23 0 0 27 27 8 8 0 0 8 18 27 30 0 0 33 33 14 14

`G:=sub<GL(6,GF(41))| [32,0,0,0,0,0,0,32,0,0,0,0,0,0,30,0,32,0,0,0,0,30,0,32,0,0,9,0,11,0,0,0,0,9,0,11],[32,0,0,0,0,0,0,32,0,0,0,0,0,0,24,1,0,0,0,0,40,17,0,0,0,0,0,0,24,1,0,0,0,0,40,17],[17,21,0,0,0,0,39,24,0,0,0,0,0,0,0,0,40,34,0,0,0,0,7,7,0,0,40,34,0,0,0,0,7,7,0,0],[17,22,0,0,0,0,39,24,0,0,0,0,0,0,14,27,8,33,0,0,11,27,18,33,0,0,33,8,27,14,0,0,23,8,30,14] >;`

C42.116D10 in GAP, Magma, Sage, TeX

`C_4^2._{116}D_{10}`
`% in TeX`

`G:=Group("C4^2.116D10");`
`// GroupNames label`

`G:=SmallGroup(320,1234);`
`// by ID`

`G=gap.SmallGroup(320,1234);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,758,219,675,192,12550]);`
`// Polycyclic`

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

׿
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