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## G = D10⋊2M4(2)  order 320 = 26·5

### 2nd semidirect product of D10 and M4(2) acting via M4(2)/C4=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — D10⋊2M4(2)
 Chief series C1 — C5 — C10 — Dic5 — C2×Dic5 — C2×C5⋊C8 — C2×D5⋊C8 — D10⋊2M4(2)
 Lower central C5 — C2×C10 — D10⋊2M4(2)
 Upper central C1 — C22 — C4⋊C4

Generators and relations for D102M4(2)
G = < a,b,c,d | a10=b2=c8=d2=1, bab=dad=a-1, cac-1=a7, cbc-1=ab, dbd=a3b, dcd=c5 >

Subgroups: 474 in 124 conjugacy classes, 46 normal (42 characteristic)
C1, C2 [×3], C2 [×3], C4 [×6], C22, C22 [×7], C5, C8 [×5], C2×C4 [×3], C2×C4 [×6], D4 [×2], C23 [×2], D5 [×3], C10 [×3], C42, C22⋊C4 [×2], C4⋊C4, C2×C8 [×6], M4(2) [×2], C22×C4 [×2], C2×D4, Dic5 [×2], Dic5, C20 [×3], D10 [×2], D10 [×5], C2×C10, C8⋊C4, C22⋊C8 [×2], C4⋊C8, C4×D4, C22×C8, C2×M4(2), C5⋊C8 [×2], C5⋊C8 [×3], C4×D5 [×4], D20 [×2], C2×Dic5 [×2], C2×C20 [×3], C22×D5 [×2], C89D4, C4×Dic5, D10⋊C4 [×2], C5×C4⋊C4, D5⋊C8 [×2], C4.F5 [×2], C2×C5⋊C8 [×4], C2×C4×D5 [×2], C2×D20, C10.C42, D10⋊C8 [×2], Dic5⋊C8, D208C4, C2×D5⋊C8, C2×C4.F5, D102M4(2)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], C23, M4(2) [×2], C22×C4, C2×D4, C4○D4, F5, C4×D4, C2×M4(2), C8○D4, C2×F5 [×3], C89D4, C22×F5, D5⋊M4(2), D4×F5, Q8.F5, D102M4(2)

Smallest permutation representation of D102M4(2)
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 77)(2 76)(3 75)(4 74)(5 73)(6 72)(7 71)(8 80)(9 79)(10 78)(11 107)(12 106)(13 105)(14 104)(15 103)(16 102)(17 101)(18 110)(19 109)(20 108)(21 85)(22 84)(23 83)(24 82)(25 81)(26 90)(27 89)(28 88)(29 87)(30 86)(31 67)(32 66)(33 65)(34 64)(35 63)(36 62)(37 61)(38 70)(39 69)(40 68)(41 54)(42 53)(43 52)(44 51)(45 60)(46 59)(47 58)(48 57)(49 56)(50 55)(91 158)(92 157)(93 156)(94 155)(95 154)(96 153)(97 152)(98 151)(99 160)(100 159)(111 147)(112 146)(113 145)(114 144)(115 143)(116 142)(117 141)(118 150)(119 149)(120 148)(121 134)(122 133)(123 132)(124 131)(125 140)(126 139)(127 138)(128 137)(129 136)(130 135)
(1 114 38 105 21 130 44 96)(2 117 37 102 22 123 43 93)(3 120 36 109 23 126 42 100)(4 113 35 106 24 129 41 97)(5 116 34 103 25 122 50 94)(6 119 33 110 26 125 49 91)(7 112 32 107 27 128 48 98)(8 115 31 104 28 121 47 95)(9 118 40 101 29 124 46 92)(10 111 39 108 30 127 45 99)(11 82 139 56 151 74 148 65)(12 85 138 53 152 77 147 62)(13 88 137 60 153 80 146 69)(14 81 136 57 154 73 145 66)(15 84 135 54 155 76 144 63)(16 87 134 51 156 79 143 70)(17 90 133 58 157 72 142 67)(18 83 132 55 158 75 141 64)(19 86 131 52 159 78 150 61)(20 89 140 59 160 71 149 68)
(2 10)(3 9)(4 8)(5 7)(11 152)(12 151)(13 160)(14 159)(15 158)(16 157)(17 156)(18 155)(19 154)(20 153)(22 30)(23 29)(24 28)(25 27)(31 35)(32 34)(36 40)(37 39)(41 47)(42 46)(43 45)(48 50)(51 58)(52 57)(53 56)(54 55)(59 60)(61 66)(62 65)(63 64)(67 70)(68 69)(71 80)(72 79)(73 78)(74 77)(75 76)(81 86)(82 85)(83 84)(87 90)(88 89)(91 110)(92 109)(93 108)(94 107)(95 106)(96 105)(97 104)(98 103)(99 102)(100 101)(111 123)(112 122)(113 121)(114 130)(115 129)(116 128)(117 127)(118 126)(119 125)(120 124)(131 145)(132 144)(133 143)(134 142)(135 141)(136 150)(137 149)(138 148)(139 147)(140 146)```

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

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

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

38 conjugacy classes

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

38 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 4 4 4 8 8 type + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C4 C4 C4 D4 C4○D4 M4(2) C8○D4 F5 C2×F5 D5⋊M4(2) D4×F5 Q8.F5 kernel D10⋊2M4(2) C10.C42 D10⋊C8 Dic5⋊C8 D20⋊8C4 C2×D5⋊C8 C2×C4.F5 D10⋊C4 C5×C4⋊C4 C2×D20 C5⋊C8 Dic5 D10 C10 C4⋊C4 C2×C4 C2 C2 C2 # reps 1 1 2 1 1 1 1 4 2 2 2 2 4 4 1 3 4 1 1

Matrix representation of D102M4(2) in GL6(𝔽41)

 40 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 1 0 0 0 0 40 0 0 0 1 0 40 0 0 0 0 1 40 0
,
 0 40 0 0 0 0 40 0 0 0 0 0 0 0 38 22 19 3 0 0 19 0 22 3 0 0 38 3 22 0 0 0 0 3 19 22
,
 40 0 0 0 0 0 0 1 0 0 0 0 0 0 1 9 11 11 0 0 12 18 12 20 0 0 21 29 23 29 0 0 30 30 32 40
,
 1 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0

`G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,40,40,40,40,0,0,1,0,0,0],[0,40,0,0,0,0,40,0,0,0,0,0,0,0,38,19,38,0,0,0,22,0,3,3,0,0,19,22,22,19,0,0,3,3,0,22],[40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,12,21,30,0,0,9,18,29,30,0,0,11,12,23,32,0,0,11,20,29,40],[1,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1,0,0,0] >;`

D102M4(2) in GAP, Magma, Sage, TeX

`D_{10}\rtimes_2M_4(2)`
`% in TeX`

`G:=Group("D10:2M4(2)");`
`// GroupNames label`

`G:=SmallGroup(320,1042);`
`// by ID`

`G=gap.SmallGroup(320,1042);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,232,758,219,184,136,6278,1595]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^10=b^2=c^8=d^2=1,b*a*b=d*a*d=a^-1,c*a*c^-1=a^7,c*b*c^-1=a*b,d*b*d=a^3*b,d*c*d=c^5>;`
`// generators/relations`

׿
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