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## G = C42.D10order 320 = 26·5

### 1st 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.D10
 Chief series C1 — C5 — C10 — C2×C10 — C2×C20 — C4×C20 — C4.D20 — C42.D10
 Lower central C5 — C2×C10 — C2×C20 — C42.D10
 Upper central C1 — C22 — C42 — C8⋊C4

Generators and relations for C42.D10
G = < a,b,c,d | a4=b4=1, c10=a, d2=a-1b, ab=ba, ac=ca, dad-1=a-1b2, cbc-1=a2b, dbd-1=a2b-1, dcd-1=a2bc9 >

Subgroups: 350 in 70 conjugacy classes, 25 normal (all characteristic)
C1, C2, C2, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C42, C22⋊C4, C2×C8, C2×D4, C2×Q8, Dic5, C20, D10, C2×C10, C8⋊C4, C8⋊C4, C4.4D4, C52C8, C40, Dic10, D20, C2×Dic5, C2×C20, C22×D5, C42.C22, C2×C52C8, D10⋊C4, C4×C20, C2×C40, C2×Dic10, C2×D20, C42.D5, C5×C8⋊C4, C4.D20, C42.D10
Quotients: C1, C2, C4, C22, C2×C4, D4, D5, C22⋊C4, D10, C4.D4, C4≀C2, C4×D5, D20, C5⋊D4, C42.C22, D10⋊C4, D204C4, C20.46D4, D207C4, C42.D10

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

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

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

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

59 conjugacy classes

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

59 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + image C1 C2 C2 C2 C4 C4 D4 D5 D10 C4≀C2 C4×D5 D20 C5⋊D4 D20⋊4C4 C4.D4 C20.46D4 D20⋊7C4 kernel C42.D10 C42.D5 C5×C8⋊C4 C4.D20 C2×Dic10 C2×D20 C2×C20 C8⋊C4 C42 C10 C2×C4 C2×C4 C2×C4 C2 C10 C2 C2 # reps 1 1 1 1 2 2 2 2 2 8 4 4 4 16 1 4 4

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

 24 35 0 0 7 17 0 0 0 0 32 0 0 0 0 32
,
 30 13 0 0 19 11 0 0 0 0 0 1 0 0 40 0
,
 36 6 0 0 34 2 0 0 0 0 4 4 0 0 4 37
,
 2 35 0 0 9 39 0 0 0 0 4 37 0 0 4 4
`G:=sub<GL(4,GF(41))| [24,7,0,0,35,17,0,0,0,0,32,0,0,0,0,32],[30,19,0,0,13,11,0,0,0,0,0,40,0,0,1,0],[36,34,0,0,6,2,0,0,0,0,4,4,0,0,4,37],[2,9,0,0,35,39,0,0,0,0,4,4,0,0,37,4] >;`

C42.D10 in GAP, Magma, Sage, TeX

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

`G:=Group("C4^2.D10");`
`// GroupNames label`

`G:=SmallGroup(320,22);`
`// by ID`

`G=gap.SmallGroup(320,22);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,141,36,422,184,1571,570,192,12550]);`
`// Polycyclic`

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

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