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## G = C42.19D10order 320 = 26·5

### 19th non-split extension by C42 of D10 acting via D10/C5=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — C42.19D10
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C2×D20 — C20⋊4D4 — C42.19D10
 Lower central C5 — C10 — C2×C20 — C42.19D10
 Upper central C1 — C22 — C42 — C8⋊C4

Generators and relations for C42.19D10
G = < a,b,c,d | a4=b4=1, c10=a2b, d2=a2b2, ab=ba, cac-1=ab2, dad-1=a-1b2, bc=cb, dbd-1=b-1, dcd-1=bc9 >

Subgroups: 638 in 110 conjugacy classes, 39 normal (15 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, C2×C4, D4, C23, D5, C10, C10, C42, C4⋊C4, C2×C8, C2×D4, Dic5, C20, C20, D10, C2×C10, C8⋊C4, D4⋊C4, C42.C2, C41D4, C40, D20, C2×Dic5, C2×C20, C2×C20, C22×D5, C42.29C22, C10.D4, C4⋊Dic5, C4×C20, C2×C40, C2×D20, C2×D20, D205C4, C5×C8⋊C4, C20.6Q8, C204D4, C42.19D10
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, D10, C4.4D4, C8⋊C22, D20, C22×D5, C42.29C22, C2×D20, C4○D20, C4.D20, C8⋊D10, C42.19D10

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

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

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

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

56 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 5A 5B 8A 8B 8C 8D 10A ··· 10F 20A ··· 20H 20I ··· 20P 40A ··· 40P order 1 2 2 2 2 2 4 4 4 4 4 4 5 5 8 8 8 8 10 ··· 10 20 ··· 20 20 ··· 20 40 ··· 40 size 1 1 1 1 40 40 2 2 4 4 40 40 2 2 4 4 4 4 2 ··· 2 2 ··· 2 4 ··· 4 4 ··· 4

56 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + image C1 C2 C2 C2 C2 D4 D5 C4○D4 D10 D10 D20 C4○D20 C8⋊C22 C8⋊D10 kernel C42.19D10 D20⋊5C4 C5×C8⋊C4 C20.6Q8 C20⋊4D4 C2×C20 C8⋊C4 C20 C42 C2×C8 C2×C4 C4 C10 C2 # reps 1 4 1 1 1 2 2 4 2 4 8 16 2 8

Matrix representation of C42.19D10 in GL6(𝔽41)

 1 2 0 0 0 0 40 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 30 13 0 0 0 0 19 11 0 0 0 0 0 0 30 13 0 0 0 0 19 11
,
 9 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 0 35 0 0 0 0 7 34 0 0 9 16 0 0 0 0 36 14 0 0
,
 9 0 0 0 0 0 32 32 0 0 0 0 0 0 0 0 14 25 0 0 0 0 25 27 0 0 27 16 0 0 0 0 16 14 0 0

`G:=sub<GL(6,GF(41))| [1,40,0,0,0,0,2,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,30,19,0,0,0,0,13,11,0,0,0,0,0,0,30,19,0,0,0,0,13,11],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,9,36,0,0,0,0,16,14,0,0,0,7,0,0,0,0,35,34,0,0],[9,32,0,0,0,0,0,32,0,0,0,0,0,0,0,0,27,16,0,0,0,0,16,14,0,0,14,25,0,0,0,0,25,27,0,0] >;`

C42.19D10 in GAP, Magma, Sage, TeX

`C_4^2._{19}D_{10}`
`% in TeX`

`G:=Group("C4^2.19D10");`
`// GroupNames label`

`G:=SmallGroup(320,340);`
`// by ID`

`G=gap.SmallGroup(320,340);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,120,254,387,142,1123,136,12550]);`
`// Polycyclic`

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

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