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## G = C42.119D10order 320 = 26·5

### 119th non-split extension by C42 of D10 acting via D10/C5=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C42.119D10
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C2×C4×D5 — D10.12D4 — C42.119D10
 Lower central C5 — C2×C10 — C42.119D10
 Upper central C1 — C22 — C4×D4

Generators and relations for C42.119D10
G = < a,b,c,d | a4=b4=c10=1, d2=a2, ab=ba, cac-1=dad-1=a-1b2, bc=cb, dbd-1=b-1, dcd-1=a2c-1 >

Subgroups: 838 in 238 conjugacy classes, 99 normal (51 characteristic)
C1, C2 [×3], C2 [×5], C4 [×2], C4 [×10], C22, C22 [×2], C22 [×11], C5, C2×C4 [×5], C2×C4 [×14], D4 [×10], C23 [×2], C23 [×2], D5 [×2], C10 [×3], C10 [×3], C42, C42 [×2], C22⋊C4 [×2], C22⋊C4 [×8], C4⋊C4, C4⋊C4 [×9], C22×C4 [×2], C22×C4 [×4], C2×D4, C2×D4 [×5], Dic5 [×6], C20 [×2], C20 [×4], D10 [×6], C2×C10, C2×C10 [×2], C2×C10 [×5], C2×C4⋊C4, C42⋊C2, C4×D4, C4×D4 [×3], C4⋊D4 [×4], C22.D4 [×2], C42.C2, C422C2 [×2], C4×D5 [×2], D20 [×2], C2×Dic5 [×6], C2×Dic5 [×2], C5⋊D4 [×6], C2×C20 [×5], C2×C20 [×4], C5×D4 [×2], C22×D5 [×2], C22×C10 [×2], C22.47C24, C4×Dic5 [×2], C10.D4 [×4], C4⋊Dic5 [×3], C4⋊Dic5 [×2], D10⋊C4 [×6], C23.D5 [×2], C4×C20, C5×C22⋊C4 [×2], C5×C4⋊C4, C2×C4×D5 [×2], C2×D20, C22×Dic5 [×2], C2×C5⋊D4 [×4], C22×C20 [×2], D4×C10, C20.6Q8, C4×D20, Dic54D4 [×2], D10.12D4 [×2], C4⋊C4⋊D5 [×2], C2×C4⋊Dic5, C23.21D10, C207D4 [×2], Dic5⋊D4 [×2], D4×C20, C42.119D10
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], D5, C4○D4 [×4], C24, D10 [×7], C2×C4○D4 [×2], 2+ 1+4, C22×D5 [×7], C22.47C24, C4○D20 [×2], D42D5 [×2], C23×D5, C2×C4○D20, C2×D42D5, D48D10, C42.119D10

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

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

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

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

65 conjugacy classes

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

65 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + + + + - + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 D5 C4○D4 C4○D4 D10 D10 D10 D10 D10 C4○D20 2+ 1+4 D4⋊2D5 D4⋊8D10 kernel C42.119D10 C20.6Q8 C4×D20 Dic5⋊4D4 D10.12D4 C4⋊C4⋊D5 C2×C4⋊Dic5 C23.21D10 C20⋊7D4 Dic5⋊D4 D4×C20 C4×D4 C20 C2×C10 C42 C22⋊C4 C4⋊C4 C22×C4 C2×D4 C22 C10 C4 C2 # reps 1 1 1 2 2 2 1 1 2 2 1 2 4 4 2 4 2 4 2 16 1 4 4

Matrix representation of C42.119D10 in GL4(𝔽41) generated by

 9 0 0 0 0 9 0 0 0 0 9 0 0 0 0 32
,
 39 28 0 0 13 2 0 0 0 0 1 0 0 0 0 1
,
 21 20 0 0 21 18 0 0 0 0 0 1 0 0 1 0
,
 21 20 0 0 23 20 0 0 0 0 0 40 0 0 1 0
`G:=sub<GL(4,GF(41))| [9,0,0,0,0,9,0,0,0,0,9,0,0,0,0,32],[39,13,0,0,28,2,0,0,0,0,1,0,0,0,0,1],[21,21,0,0,20,18,0,0,0,0,0,1,0,0,1,0],[21,23,0,0,20,20,0,0,0,0,0,1,0,0,40,0] >;`

C42.119D10 in GAP, Magma, Sage, TeX

`C_4^2._{119}D_{10}`
`% in TeX`

`G:=Group("C4^2.119D10");`
`// GroupNames label`

`G:=SmallGroup(320,1237);`
`// by ID`

`G=gap.SmallGroup(320,1237);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,758,219,1571,192,12550]);`
`// 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=b^-1,d*c*d^-1=a^2*c^-1>;`
`// generators/relations`

׿
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