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## G = D10⋊M4(2)  order 320 = 26·5

### 1st 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⋊M4(2)
 Chief series C1 — C5 — C10 — Dic5 — C2×Dic5 — C2×C5⋊C8 — C2×D5⋊C8 — D10⋊M4(2)
 Lower central C5 — C2×C10 — D10⋊M4(2)
 Upper central C1 — C22 — C22⋊C4

Generators and relations for D10⋊M4(2)
G = < a,b,c,d | a10=b2=c8=d2=1, bab=a-1, cac-1=a3, ad=da, cbc-1=a7b, dbd=a5b, dcd=c5 >

Subgroups: 442 in 124 conjugacy classes, 46 normal (42 characteristic)
C1, C2 [×3], C2 [×3], C4 [×6], C22, C22 [×7], C5, C8 [×5], C2×C4 [×2], C2×C4 [×7], D4 [×2], C23, C23, D5 [×2], C10 [×3], C10, C42, C22⋊C4, C22⋊C4, C4⋊C4, C2×C8 [×6], M4(2) [×2], C22×C4 [×2], C2×D4, Dic5 [×2], Dic5 [×2], C20 [×2], D10 [×2], D10 [×2], C2×C10, C2×C10 [×3], C8⋊C4, C22⋊C8 [×2], C4⋊C8, C4×D4, C22×C8, C2×M4(2), C5⋊C8 [×2], C5⋊C8 [×3], C4×D5 [×2], C2×Dic5 [×3], C2×Dic5 [×2], C5⋊D4 [×2], C2×C20 [×2], C22×D5, C22×C10, C89D4, C4×Dic5, C10.D4, D10⋊C4, C5×C22⋊C4, D5⋊C8 [×2], C2×C5⋊C8 [×4], C22.F5 [×2], C2×C4×D5, C22×Dic5, C2×C5⋊D4, C20⋊C8, C10.C42, D10⋊C8, C23.2F5, Dic54D4, C2×D5⋊C8, C2×C22.F5, D10⋊M4(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, D4×F5, D10⋊M4(2)

Smallest permutation representation of D10⋊M4(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 10)(2 9)(3 8)(4 7)(5 6)(12 20)(13 19)(14 18)(15 17)(21 30)(22 29)(23 28)(24 27)(25 26)(31 34)(32 33)(35 40)(36 39)(37 38)(41 46)(42 45)(43 44)(47 50)(48 49)(51 57)(52 56)(53 55)(58 60)(61 65)(62 64)(66 70)(67 69)(71 79)(72 78)(73 77)(74 76)(81 85)(82 84)(86 90)(87 89)(91 100)(92 99)(93 98)(94 97)(95 96)(101 108)(102 107)(103 106)(104 105)(109 110)(111 116)(112 115)(113 114)(117 120)(118 119)(121 128)(122 127)(123 126)(124 125)(129 130)(131 135)(132 134)(136 140)(137 139)(141 143)(144 150)(145 149)(146 148)(152 160)(153 159)(154 158)(155 157)
(1 154 38 145 21 14 44 136)(2 151 37 148 22 11 43 139)(3 158 36 141 23 18 42 132)(4 155 35 144 24 15 41 135)(5 152 34 147 25 12 50 138)(6 159 33 150 26 19 49 131)(7 156 32 143 27 16 48 134)(8 153 31 146 28 13 47 137)(9 160 40 149 29 20 46 140)(10 157 39 142 30 17 45 133)(51 117 79 102 70 123 87 93)(52 114 78 105 61 130 86 96)(53 111 77 108 62 127 85 99)(54 118 76 101 63 124 84 92)(55 115 75 104 64 121 83 95)(56 112 74 107 65 128 82 98)(57 119 73 110 66 125 81 91)(58 116 72 103 67 122 90 94)(59 113 71 106 68 129 89 97)(60 120 80 109 69 126 88 100)
(1 52)(2 53)(3 54)(4 55)(5 56)(6 57)(7 58)(8 59)(9 60)(10 51)(11 111)(12 112)(13 113)(14 114)(15 115)(16 116)(17 117)(18 118)(19 119)(20 120)(21 61)(22 62)(23 63)(24 64)(25 65)(26 66)(27 67)(28 68)(29 69)(30 70)(31 71)(32 72)(33 73)(34 74)(35 75)(36 76)(37 77)(38 78)(39 79)(40 80)(41 83)(42 84)(43 85)(44 86)(45 87)(46 88)(47 89)(48 90)(49 81)(50 82)(91 150)(92 141)(93 142)(94 143)(95 144)(96 145)(97 146)(98 147)(99 148)(100 149)(101 132)(102 133)(103 134)(104 135)(105 136)(106 137)(107 138)(108 139)(109 140)(110 131)(121 155)(122 156)(123 157)(124 158)(125 159)(126 160)(127 151)(128 152)(129 153)(130 154)```

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

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

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

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 10D 10E 20A 20B 20C 20D 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 10 10 20 20 20 20 size 1 1 1 1 4 10 10 2 2 4 5 5 5 5 20 20 4 10 ··· 10 20 20 20 20 4 4 4 8 8 8 8 8 8

38 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 4 4 4 4 8 8 type + + + + + + + + + + + + - + image C1 C2 C2 C2 C2 C2 C2 C2 C4 C4 C4 C4 D4 C4○D4 M4(2) C8○D4 F5 C2×F5 C2×F5 D5⋊M4(2) D4.F5 D4×F5 kernel D10⋊M4(2) C20⋊C8 C10.C42 D10⋊C8 C23.2F5 Dic5⋊4D4 C2×D5⋊C8 C2×C22.F5 C10.D4 D10⋊C4 C5×C22⋊C4 C2×C5⋊D4 C5⋊C8 Dic5 D10 C10 C22⋊C4 C2×C4 C23 C2 C2 C2 # reps 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 1 2 1 4 1 1

Matrix representation of D10⋊M4(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
,
 40 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 40 0 0 0 1 0 40 0 0 0 0 0 40 0 0 0 0 0 40 1
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 24 17 2 0 0 0 26 17 0 24 0 0 24 0 17 26 0 0 0 2 17 24
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 22 38 0 3 0 0 0 19 38 3 0 0 3 38 19 0 0 0 3 0 38 22

`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],[40,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,40,40,40,40,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,24,26,24,0,0,0,17,17,0,2,0,0,2,0,17,17,0,0,0,24,26,24],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,22,0,3,3,0,0,38,19,38,0,0,0,0,38,19,38,0,0,3,3,0,22] >;`

D10⋊M4(2) in GAP, Magma, Sage, TeX

`D_{10}\rtimes M_4(2)`
`% in TeX`

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

`G:=SmallGroup(320,1032);`
`// by ID`

`G=gap.SmallGroup(320,1032);`
`# by ID`

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

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

׿
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