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## G = D10⋊2C42order 320 = 26·5

### 1st semidirect product of D10 and C42 acting via C42/C2×C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — D10⋊2C42
 Chief series C1 — C5 — C10 — C2×C10 — C22×C10 — C23×D5 — D5×C22×C4 — D10⋊2C42
 Lower central C5 — C10 — D10⋊2C42
 Upper central C1 — C23 — C2.C42

Generators and relations for D102C42
G = < a,b,c,d | a10=b2=c4=d4=1, bab=cac-1=a-1, ad=da, cbc-1=a8b, dbd-1=a5b, cd=dc >

Subgroups: 910 in 258 conjugacy classes, 99 normal (25 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C5, C2×C4, C2×C4, C23, C23, D5, C10, C10, C42, C22⋊C4, C22×C4, C22×C4, C22×C4, C24, Dic5, Dic5, C20, D10, D10, C2×C10, C2×C10, C2.C42, C2.C42, C2×C42, C2×C22⋊C4, C23×C4, C4×D5, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×D5, C22×D5, C22×C10, C4×C22⋊C4, C4×Dic5, D10⋊C4, C2×C4×D5, C2×C4×D5, C22×Dic5, C22×Dic5, C22×C20, C22×C20, C23×D5, C10.10C42, C5×C2.C42, C2×C4×Dic5, C2×D10⋊C4, D5×C22×C4, D102C42
Quotients:

Smallest permutation representation of D102C42
On 160 points
Generators in S160
```(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)(2 9)(3 8)(4 7)(5 6)(11 14)(12 13)(15 20)(16 19)(17 18)(21 24)(22 23)(25 30)(26 29)(27 28)(31 34)(32 33)(35 40)(36 39)(37 38)(41 49)(42 48)(43 47)(44 46)(51 59)(52 58)(53 57)(54 56)(61 69)(62 68)(63 67)(64 66)(71 79)(72 78)(73 77)(74 76)(81 84)(82 83)(85 90)(86 89)(87 88)(91 94)(92 93)(95 100)(96 99)(97 98)(101 104)(102 103)(105 110)(106 109)(107 108)(111 114)(112 113)(115 120)(116 119)(117 118)(121 129)(122 128)(123 127)(124 126)(131 139)(132 138)(133 137)(134 136)(141 149)(142 148)(143 147)(144 146)(151 159)(152 158)(153 157)(154 156)
(1 93 13 83)(2 92 14 82)(3 91 15 81)(4 100 16 90)(5 99 17 89)(6 98 18 88)(7 97 19 87)(8 96 20 86)(9 95 11 85)(10 94 12 84)(21 115 31 105)(22 114 32 104)(23 113 33 103)(24 112 34 102)(25 111 35 101)(26 120 36 110)(27 119 37 109)(28 118 38 108)(29 117 39 107)(30 116 40 106)(41 140 51 130)(42 139 52 129)(43 138 53 128)(44 137 54 127)(45 136 55 126)(46 135 56 125)(47 134 57 124)(48 133 58 123)(49 132 59 122)(50 131 60 121)(61 160 71 150)(62 159 72 149)(63 158 73 148)(64 157 74 147)(65 156 75 146)(66 155 76 145)(67 154 77 144)(68 153 78 143)(69 152 79 142)(70 151 80 141)
(1 63 23 43)(2 64 24 44)(3 65 25 45)(4 66 26 46)(5 67 27 47)(6 68 28 48)(7 69 29 49)(8 70 30 50)(9 61 21 41)(10 62 22 42)(11 71 31 51)(12 72 32 52)(13 73 33 53)(14 74 34 54)(15 75 35 55)(16 76 36 56)(17 77 37 57)(18 78 38 58)(19 79 39 59)(20 80 40 60)(81 146 101 126)(82 147 102 127)(83 148 103 128)(84 149 104 129)(85 150 105 130)(86 141 106 121)(87 142 107 122)(88 143 108 123)(89 144 109 124)(90 145 110 125)(91 156 111 136)(92 157 112 137)(93 158 113 138)(94 159 114 139)(95 160 115 140)(96 151 116 131)(97 152 117 132)(98 153 118 133)(99 154 119 134)(100 155 120 135)```

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

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

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

80 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 4A ··· 4L 4M ··· 4T 4U ··· 4AB 5A 5B 10A ··· 10N 20A ··· 20X order 1 2 ··· 2 2 2 2 2 4 ··· 4 4 ··· 4 4 ··· 4 5 5 10 ··· 10 20 ··· 20 size 1 1 ··· 1 10 10 10 10 2 ··· 2 5 ··· 5 10 ··· 10 2 2 2 ··· 2 4 ··· 4

80 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 4 4 4 type + + + + + + + + + + - + image C1 C2 C2 C2 C2 C2 C4 C4 D4 D5 C4○D4 D10 C4×D5 D4×D5 D4⋊2D5 Q8⋊2D5 kernel D10⋊2C42 C10.10C42 C5×C2.C42 C2×C4×Dic5 C2×D10⋊C4 D5×C22×C4 D10⋊C4 C2×C4×D5 C2×Dic5 C2.C42 C2×C10 C22×C4 C2×C4 C22 C22 C22 # reps 1 1 1 2 2 1 16 8 4 2 4 6 24 4 2 2

Matrix representation of D102C42 in GL6(𝔽41)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 35 35 0 0 0 0 6 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 35 35 0 0 0 0 40 6 0 0 0 0 0 0 40 0 0 0 0 0 1 1
,
 32 0 0 0 0 0 0 32 0 0 0 0 0 0 40 0 0 0 0 0 35 1 0 0 0 0 0 0 32 0 0 0 0 0 0 32
,
 1 0 0 0 0 0 0 9 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 40 39 0 0 0 0 1 1

`G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,35,6,0,0,0,0,35,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,35,40,0,0,0,0,35,6,0,0,0,0,0,0,40,1,0,0,0,0,0,1],[32,0,0,0,0,0,0,32,0,0,0,0,0,0,40,35,0,0,0,0,0,1,0,0,0,0,0,0,32,0,0,0,0,0,0,32],[1,0,0,0,0,0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,1,0,0,0,0,39,1] >;`

D102C42 in GAP, Magma, Sage, TeX

`D_{10}\rtimes_2C_4^2`
`% in TeX`

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

`G:=SmallGroup(320,293);`
`// by ID`

`G=gap.SmallGroup(320,293);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,758,387,58,12550]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^10=b^2=c^4=d^4=1,b*a*b=c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=a^8*b,d*b*d^-1=a^5*b,c*d=d*c>;`
`// generators/relations`

׿
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