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## G = C42.D5order 160 = 25·5

### 1st non-split extension by C42 of D5 acting via D5/C5=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C42.D5
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C2×C5⋊2C8 — C42.D5
 Lower central C5 — C10 — C42.D5
 Upper central C1 — C2×C4 — C42

Generators and relations for C42.D5
G = < a,b,c,d | a4=b4=c5=1, d2=b, ab=ba, ac=ca, dad-1=ab2, bc=cb, bd=db, dcd-1=c-1 >

Smallest permutation representation of C42.D5
Regular action on 160 points
Generators in S160
```(1 109 159 44)(2 106 160 41)(3 111 153 46)(4 108 154 43)(5 105 155 48)(6 110 156 45)(7 107 157 42)(8 112 158 47)(9 61 86 136)(10 58 87 133)(11 63 88 130)(12 60 81 135)(13 57 82 132)(14 62 83 129)(15 59 84 134)(16 64 85 131)(17 140 148 36)(18 137 149 33)(19 142 150 38)(20 139 151 35)(21 144 152 40)(22 141 145 37)(23 138 146 34)(24 143 147 39)(25 55 65 114)(26 52 66 119)(27 49 67 116)(28 54 68 113)(29 51 69 118)(30 56 70 115)(31 53 71 120)(32 50 72 117)(73 122 103 95)(74 127 104 92)(75 124 97 89)(76 121 98 94)(77 126 99 91)(78 123 100 96)(79 128 101 93)(80 125 102 90)
(1 3 5 7)(2 4 6 8)(9 11 13 15)(10 12 14 16)(17 19 21 23)(18 20 22 24)(25 27 29 31)(26 28 30 32)(33 35 37 39)(34 36 38 40)(41 43 45 47)(42 44 46 48)(49 51 53 55)(50 52 54 56)(57 59 61 63)(58 60 62 64)(65 67 69 71)(66 68 70 72)(73 75 77 79)(74 76 78 80)(81 83 85 87)(82 84 86 88)(89 91 93 95)(90 92 94 96)(97 99 101 103)(98 100 102 104)(105 107 109 111)(106 108 110 112)(113 115 117 119)(114 116 118 120)(121 123 125 127)(122 124 126 128)(129 131 133 135)(130 132 134 136)(137 139 141 143)(138 140 142 144)(145 147 149 151)(146 148 150 152)(153 155 157 159)(154 156 158 160)
(1 51 132 101 33)(2 34 102 133 52)(3 53 134 103 35)(4 36 104 135 54)(5 55 136 97 37)(6 38 98 129 56)(7 49 130 99 39)(8 40 100 131 50)(9 89 22 105 65)(10 66 106 23 90)(11 91 24 107 67)(12 68 108 17 92)(13 93 18 109 69)(14 70 110 19 94)(15 95 20 111 71)(16 72 112 21 96)(25 86 124 145 48)(26 41 146 125 87)(27 88 126 147 42)(28 43 148 127 81)(29 82 128 149 44)(30 45 150 121 83)(31 84 122 151 46)(32 47 152 123 85)(57 79 137 159 118)(58 119 160 138 80)(59 73 139 153 120)(60 113 154 140 74)(61 75 141 155 114)(62 115 156 142 76)(63 77 143 157 116)(64 117 158 144 78)
(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)```

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

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

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

52 conjugacy classes

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

52 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 type + + + + - + image C1 C2 C2 C4 C4 D5 M4(2) Dic5 D10 C4×D5 C4.Dic5 kernel C42.D5 C2×C5⋊2C8 C4×C20 C5⋊2C8 C2×C20 C42 C10 C2×C4 C2×C4 C4 C2 # reps 1 2 1 8 4 2 4 4 2 8 16

Matrix representation of C42.D5 in GL4(𝔽41) generated by

 9 0 0 0 0 9 0 0 0 0 18 35 0 0 6 23
,
 1 0 0 0 0 1 0 0 0 0 32 0 0 0 0 32
,
 40 1 0 0 33 7 0 0 0 0 0 1 0 0 40 6
,
 40 1 0 0 0 1 0 0 0 0 37 35 0 0 11 4
`G:=sub<GL(4,GF(41))| [9,0,0,0,0,9,0,0,0,0,18,6,0,0,35,23],[1,0,0,0,0,1,0,0,0,0,32,0,0,0,0,32],[40,33,0,0,1,7,0,0,0,0,0,40,0,0,1,6],[40,0,0,0,1,1,0,0,0,0,37,11,0,0,35,4] >;`

C42.D5 in GAP, Magma, Sage, TeX

`C_4^2.D_5`
`% in TeX`

`G:=Group("C4^2.D5");`
`// GroupNames label`

`G:=SmallGroup(160,10);`
`// by ID`

`G=gap.SmallGroup(160,10);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-5,24,217,55,86,4613]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^4=b^4=c^5=1,d^2=b,a*b=b*a,a*c=c*a,d*a*d^-1=a*b^2,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;`
`// generators/relations`

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