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G = C42.159D10order 320 = 26·5

159th non-split extension by C42 of D10 acting via D10/C5=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C42.159D10
 Chief series C1 — C5 — C10 — C2×C10 — C2×Dic5 — C4×Dic5 — Dic5⋊3Q8 — C42.159D10
 Lower central C5 — C2×C10 — C42.159D10
 Upper central C1 — C22 — C42⋊2C2

Generators and relations for C42.159D10
G = < a,b,c,d | a4=b4=1, c10=b2, d2=a2, ab=ba, cac-1=dad-1=ab2, cbc-1=a2b-1, dbd-1=a2b, dcd-1=c9 >

Subgroups: 558 in 192 conjugacy classes, 93 normal (91 characteristic)
C1, C2 [×3], C2, C4 [×15], C22, C22 [×3], C5, C2×C4 [×6], C2×C4 [×10], Q8 [×4], C23, C10 [×3], C10, C42, C42 [×5], C22⋊C4 [×3], C22⋊C4 [×3], C4⋊C4 [×3], C4⋊C4 [×17], C22×C4, C2×Q8 [×2], Dic5 [×2], Dic5 [×7], C20 [×6], C2×C10, C2×C10 [×3], C42⋊C2, C4×Q8 [×2], C22⋊Q8 [×2], C42.C2 [×5], C422C2, C422C2 [×3], C4⋊Q8, Dic10 [×4], C2×Dic5 [×8], C2×Dic5 [×2], C2×C20 [×6], C22×C10, C22.35C24, C4×Dic5 [×5], C10.D4 [×12], C4⋊Dic5 [×5], C23.D5 [×3], C4×C20, C5×C22⋊C4 [×3], C5×C4⋊C4 [×3], C2×Dic10 [×2], C22×Dic5, C4×Dic10, C20.6Q8, C23.11D10, Dic5.14D4 [×2], C23.D10 [×3], Dic53Q8, C20⋊Q8, Dic5.Q8 [×3], C4.Dic10, C5×C422C2, C42.159D10
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], C22.35C24, C23×D5, D5×C4○D4, D4.10D10 [×2], C42.159D10

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

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

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

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

50 conjugacy classes

 class 1 2A 2B 2C 2D 4A 4B 4C ··· 4G 4H 4I 4J 4K 4L ··· 4Q 5A 5B 10A ··· 10F 10G 10H 20A ··· 20L 20M ··· 20R order 1 2 2 2 2 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 4 2 2 4 ··· 4 10 10 10 10 20 ··· 20 2 2 2 ··· 2 8 8 4 ··· 4 8 ··· 8

50 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + - - image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 D5 C4○D4 D10 D10 D10 2- 1+4 D5×C4○D4 D4.10D10 kernel C42.159D10 C4×Dic10 C20.6Q8 C23.11D10 Dic5.14D4 C23.D10 Dic5⋊3Q8 C20⋊Q8 Dic5.Q8 C4.Dic10 C5×C42⋊2C2 C42⋊2C2 Dic5 C42 C22⋊C4 C4⋊C4 C10 C2 C2 # reps 1 1 1 1 2 3 1 1 3 1 1 2 4 2 6 6 2 4 8

Matrix representation of C42.159D10 in GL6(𝔽41)

 9 0 0 0 0 0 0 9 0 0 0 0 0 0 1 0 11 14 0 0 0 1 11 11 0 0 28 24 40 0 0 0 13 28 0 40
,
 40 39 0 0 0 0 0 1 0 0 0 0 0 0 2 13 0 0 0 0 28 39 0 0 0 0 0 0 39 13 0 0 0 0 28 2
,
 32 23 0 0 0 0 9 9 0 0 0 0 0 0 16 25 0 7 0 0 16 2 34 0 0 0 0 0 39 16 0 0 0 0 25 25
,
 40 39 0 0 0 0 1 1 0 0 0 0 0 0 2 14 0 0 0 0 26 39 0 0 0 0 24 16 15 10 0 0 16 3 2 26

`G:=sub<GL(6,GF(41))| [9,0,0,0,0,0,0,9,0,0,0,0,0,0,1,0,28,13,0,0,0,1,24,28,0,0,11,11,40,0,0,0,14,11,0,40],[40,0,0,0,0,0,39,1,0,0,0,0,0,0,2,28,0,0,0,0,13,39,0,0,0,0,0,0,39,28,0,0,0,0,13,2],[32,9,0,0,0,0,23,9,0,0,0,0,0,0,16,16,0,0,0,0,25,2,0,0,0,0,0,34,39,25,0,0,7,0,16,25],[40,1,0,0,0,0,39,1,0,0,0,0,0,0,2,26,24,16,0,0,14,39,16,3,0,0,0,0,15,2,0,0,0,0,10,26] >;`

C42.159D10 in GAP, Magma, Sage, TeX

`C_4^2._{159}D_{10}`
`% in TeX`

`G:=Group("C4^2.159D10");`
`// GroupNames label`

`G:=SmallGroup(320,1373);`
`// by ID`

`G=gap.SmallGroup(320,1373);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,120,219,268,675,570,80,12550]);`
`// Polycyclic`

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

׿
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