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## G = D10⋊4M4(2)  order 320 = 26·5

### 2nd semidirect product of D10 and M4(2) acting via M4(2)/C8=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — D10⋊4M4(2)
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C2×C4×D5 — C4×C5⋊D4 — D10⋊4M4(2)
 Lower central C5 — C2×C10 — D10⋊4M4(2)
 Upper central C1 — C2×C4 — C22⋊C8

Generators and relations for D104M4(2)
G = < a,b,c,d | a10=b2=c8=d2=1, bab=cac-1=a-1, ad=da, cbc-1=a3b, dbd=a5b, dcd=c5 >

Subgroups: 398 in 124 conjugacy classes, 51 normal (47 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, C23, C23, D5, C10, C10, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), C22×C4, C22×C4, C2×D4, Dic5, C20, C20, D10, D10, C2×C10, C2×C10, C8⋊C4, C22⋊C8, C22⋊C8, C4⋊C8, C4×D4, C22×C8, C2×M4(2), C52C8, C52C8, C40, C4×D5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C22×D5, C22×C10, C89D4, C8×D5, C2×C52C8, C4.Dic5, C4×Dic5, C10.D4, D10⋊C4, C23.D5, C2×C40, C2×C4×D5, C2×C5⋊D4, C22×C20, C20.8Q8, C408C4, D101C8, C5×C22⋊C8, D5×C2×C8, C2×C4.Dic5, C4×C5⋊D4, D104M4(2)
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D5, M4(2), C22×C4, C2×D4, C4○D4, D10, C4×D4, C2×M4(2), C8○D4, C4×D5, C22×D5, C89D4, C2×C4×D5, D4×D5, D42D5, Dic54D4, D20.3C4, D5×M4(2), D104M4(2)

Smallest permutation representation of D104M4(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 22)(2 21)(3 30)(4 29)(5 28)(6 27)(7 26)(8 25)(9 24)(10 23)(11 152)(12 151)(13 160)(14 159)(15 158)(16 157)(17 156)(18 155)(19 154)(20 153)(31 41)(32 50)(33 49)(34 48)(35 47)(36 46)(37 45)(38 44)(39 43)(40 42)(51 61)(52 70)(53 69)(54 68)(55 67)(56 66)(57 65)(58 64)(59 63)(60 62)(71 81)(72 90)(73 89)(74 88)(75 87)(76 86)(77 85)(78 84)(79 83)(80 82)(91 106)(92 105)(93 104)(94 103)(95 102)(96 101)(97 110)(98 109)(99 108)(100 107)(111 126)(112 125)(113 124)(114 123)(115 122)(116 121)(117 130)(118 129)(119 128)(120 127)(131 146)(132 145)(133 144)(134 143)(135 142)(136 141)(137 150)(138 149)(139 148)(140 147)
(1 11 45 138 23 158 38 145)(2 20 46 137 24 157 39 144)(3 19 47 136 25 156 40 143)(4 18 48 135 26 155 31 142)(5 17 49 134 27 154 32 141)(6 16 50 133 28 153 33 150)(7 15 41 132 29 152 34 149)(8 14 42 131 30 151 35 148)(9 13 43 140 21 160 36 147)(10 12 44 139 22 159 37 146)(51 102 71 122 68 95 88 115)(52 101 72 121 69 94 89 114)(53 110 73 130 70 93 90 113)(54 109 74 129 61 92 81 112)(55 108 75 128 62 91 82 111)(56 107 76 127 63 100 83 120)(57 106 77 126 64 99 84 119)(58 105 78 125 65 98 85 118)(59 104 79 124 66 97 86 117)(60 103 80 123 67 96 87 116)
(1 105)(2 106)(3 107)(4 108)(5 109)(6 110)(7 101)(8 102)(9 103)(10 104)(11 85)(12 86)(13 87)(14 88)(15 89)(16 90)(17 81)(18 82)(19 83)(20 84)(21 96)(22 97)(23 98)(24 99)(25 100)(26 91)(27 92)(28 93)(29 94)(30 95)(31 111)(32 112)(33 113)(34 114)(35 115)(36 116)(37 117)(38 118)(39 119)(40 120)(41 121)(42 122)(43 123)(44 124)(45 125)(46 126)(47 127)(48 128)(49 129)(50 130)(51 131)(52 132)(53 133)(54 134)(55 135)(56 136)(57 137)(58 138)(59 139)(60 140)(61 141)(62 142)(63 143)(64 144)(65 145)(66 146)(67 147)(68 148)(69 149)(70 150)(71 151)(72 152)(73 153)(74 154)(75 155)(76 156)(77 157)(78 158)(79 159)(80 160)```

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

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

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

68 conjugacy classes

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

68 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C2 C4 C4 C4 C4 D4 D5 C4○D4 M4(2) D10 D10 C8○D4 C4×D5 C4×D5 D20.3C4 D4×D5 D4⋊2D5 D5×M4(2) kernel D10⋊4M4(2) C20.8Q8 C40⋊8C4 D10⋊1C8 C5×C22⋊C8 D5×C2×C8 C2×C4.Dic5 C4×C5⋊D4 C10.D4 D10⋊C4 C23.D5 C2×C5⋊D4 C5⋊2C8 C22⋊C8 C20 D10 C2×C8 C22×C4 C10 C2×C4 C23 C2 C4 C4 C2 # reps 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 2 4 4 4 16 2 2 4

Matrix representation of D104M4(2) in GL4(𝔽41) generated by

 0 7 0 0 35 7 0 0 0 0 40 0 0 0 0 40
,
 7 34 0 0 1 34 0 0 0 0 40 0 0 0 22 1
,
 1 7 0 0 7 40 0 0 0 0 34 18 0 0 20 7
,
 17 7 0 0 35 24 0 0 0 0 19 39 0 0 16 22
`G:=sub<GL(4,GF(41))| [0,35,0,0,7,7,0,0,0,0,40,0,0,0,0,40],[7,1,0,0,34,34,0,0,0,0,40,22,0,0,0,1],[1,7,0,0,7,40,0,0,0,0,34,20,0,0,18,7],[17,35,0,0,7,24,0,0,0,0,19,16,0,0,39,22] >;`

D104M4(2) in GAP, Magma, Sage, TeX

`D_{10}\rtimes_4M_4(2)`
`% in TeX`

`G:=Group("D10:4M4(2)");`
`// GroupNames label`

`G:=SmallGroup(320,355);`
`// by ID`

`G=gap.SmallGroup(320,355);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,701,219,58,136,12550]);`
`// Polycyclic`

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

׿
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