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## G = D10⋊5M4(2)  order 320 = 26·5

### 3rd 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⋊5M4(2)
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C2×C4×D5 — C4×D20 — D10⋊5M4(2)
 Lower central C5 — C2×C10 — D10⋊5M4(2)
 Upper central C1 — C2×C4 — C4⋊C8

Generators and relations for D105M4(2)
G = < a,b,c,d | a10=b2=c8=d2=1, bab=dad=a-1, ac=ca, cbc-1=a5b, dbd=a3b, dcd=c5 >

Subgroups: 446 in 124 conjugacy classes, 51 normal (47 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, C23, D5, C10, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), C22×C4, C2×D4, Dic5, C20, C20, D10, D10, C2×C10, C8⋊C4, C22⋊C8, C4⋊C8, C4×D4, C22×C8, C2×M4(2), C52C8, C52C8, C40, C4×D5, D20, C2×Dic5, C2×C20, C22×D5, C89D4, C8×D5, C8⋊D5, C2×C52C8, C4⋊Dic5, D10⋊C4, C4×C20, C2×C40, C2×C4×D5, C2×D20, C42.D5, D101C8, C5×C4⋊C8, C4×D20, D5×C2×C8, C2×C8⋊D5, D105M4(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, Q82D5, D208C4, D20.3C4, D5×M4(2), D105M4(2)

Smallest permutation representation of D105M4(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)
(1 117)(2 116)(3 115)(4 114)(5 113)(6 112)(7 111)(8 120)(9 119)(10 118)(11 69)(12 68)(13 67)(14 66)(15 65)(16 64)(17 63)(18 62)(19 61)(20 70)(21 126)(22 125)(23 124)(24 123)(25 122)(26 121)(27 130)(28 129)(29 128)(30 127)(31 94)(32 93)(33 92)(34 91)(35 100)(36 99)(37 98)(38 97)(39 96)(40 95)(41 108)(42 107)(43 106)(44 105)(45 104)(46 103)(47 102)(48 101)(49 110)(50 109)(51 159)(52 158)(53 157)(54 156)(55 155)(56 154)(57 153)(58 152)(59 151)(60 160)(71 146)(72 145)(73 144)(74 143)(75 142)(76 141)(77 150)(78 149)(79 148)(80 147)(81 136)(82 135)(83 134)(84 133)(85 132)(86 131)(87 140)(88 139)(89 138)(90 137)
(1 145 45 11 23 138 38 158)(2 146 46 12 24 139 39 159)(3 147 47 13 25 140 40 160)(4 148 48 14 26 131 31 151)(5 149 49 15 27 132 32 152)(6 150 50 16 28 133 33 153)(7 141 41 17 29 134 34 154)(8 142 42 18 30 135 35 155)(9 143 43 19 21 136 36 156)(10 144 44 20 22 137 37 157)(51 111 71 108 68 128 88 91)(52 112 72 109 69 129 89 92)(53 113 73 110 70 130 90 93)(54 114 74 101 61 121 81 94)(55 115 75 102 62 122 82 95)(56 116 76 103 63 123 83 96)(57 117 77 104 64 124 84 97)(58 118 78 105 65 125 85 98)(59 119 79 106 66 126 86 99)(60 120 80 107 67 127 87 100)
(1 23)(2 22)(3 21)(4 30)(5 29)(6 28)(7 27)(8 26)(9 25)(10 24)(12 20)(13 19)(14 18)(15 17)(31 42)(32 41)(33 50)(34 49)(35 48)(36 47)(37 46)(38 45)(39 44)(40 43)(51 60)(52 59)(53 58)(54 57)(55 56)(61 64)(62 63)(65 70)(66 69)(67 68)(71 80)(72 79)(73 78)(74 77)(75 76)(81 84)(82 83)(85 90)(86 89)(87 88)(91 107)(92 106)(93 105)(94 104)(95 103)(96 102)(97 101)(98 110)(99 109)(100 108)(111 127)(112 126)(113 125)(114 124)(115 123)(116 122)(117 121)(118 130)(119 129)(120 128)(131 135)(132 134)(136 140)(137 139)(141 149)(142 148)(143 147)(144 146)(151 155)(152 154)(156 160)(157 159)```

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

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

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

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 20A ··· 20H 20I ··· 20P 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 20 ··· 20 20 ··· 20 40 ··· 40 size 1 1 1 1 10 10 20 1 1 1 1 4 4 10 10 20 2 2 2 2 2 2 4 4 10 10 10 10 20 20 2 ··· 2 2 ··· 2 4 ··· 4 4 ··· 4

68 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C4 C4 C4 D4 D5 C4○D4 M4(2) D10 D10 C8○D4 C4×D5 D20.3C4 D4×D5 Q8⋊2D5 D5×M4(2) kernel D10⋊5M4(2) C42.D5 D10⋊1C8 C5×C4⋊C8 C4×D20 D5×C2×C8 C2×C8⋊D5 C4⋊Dic5 D10⋊C4 C2×D20 C5⋊2C8 C4⋊C8 C20 D10 C42 C2×C8 C10 C2×C4 C2 C4 C4 C2 # reps 1 1 2 1 1 1 1 2 4 2 2 2 2 4 2 4 4 8 16 2 2 4

Matrix representation of D105M4(2) in GL6(𝔽41)

 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 7 0 0 0 0 34 7 0 0 0 0 0 0 40 0 0 0 0 0 0 40
,
 27 32 0 0 0 0 8 14 0 0 0 0 0 0 40 0 0 0 0 0 34 1 0 0 0 0 0 0 38 2 0 0 0 0 37 3
,
 32 3 0 0 0 0 17 9 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 3 40
,
 40 0 0 0 0 0 35 1 0 0 0 0 0 0 34 7 0 0 0 0 40 7 0 0 0 0 0 0 1 0 0 0 0 0 3 40

`G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,34,0,0,0,0,7,7,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[27,8,0,0,0,0,32,14,0,0,0,0,0,0,40,34,0,0,0,0,0,1,0,0,0,0,0,0,38,37,0,0,0,0,2,3],[32,17,0,0,0,0,3,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,3,0,0,0,0,0,40],[40,35,0,0,0,0,0,1,0,0,0,0,0,0,34,40,0,0,0,0,7,7,0,0,0,0,0,0,1,3,0,0,0,0,0,40] >;`

D105M4(2) in GAP, Magma, Sage, TeX

`D_{10}\rtimes_5M_4(2)`
`% in TeX`

`G:=Group("D10:5M4(2)");`
`// GroupNames label`

`G:=SmallGroup(320,463);`
`// by ID`

`G=gap.SmallGroup(320,463);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,701,120,219,58,136,12550]);`
`// Polycyclic`

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

׿
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