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## G = D20⋊12D4order 320 = 26·5

### 5th semidirect product of D20 and D4 acting via D4/C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — D20⋊12D4
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C2×C4×D5 — D5×C4⋊C4 — D20⋊12D4
 Lower central C5 — C2×C10 — D20⋊12D4
 Upper central C1 — C22 — C4⋊Q8

Generators and relations for D2012D4
G = < a,b,c,d | a20=b2=c4=d2=1, bab=a-1, ac=ca, dad=a9, bc=cb, dbd=a18b, dcd=c-1 >

Subgroups: 1110 in 292 conjugacy classes, 107 normal (27 characteristic)
C1, C2 [×3], C2 [×6], C4 [×4], C4 [×9], C22, C22 [×14], C5, C2×C4 [×3], C2×C4 [×4], C2×C4 [×20], D4 [×14], Q8 [×4], C23 [×4], D5 [×6], C10 [×3], C42, C22⋊C4 [×8], C4⋊C4 [×4], C4⋊C4 [×6], C22×C4 [×8], C2×D4 [×6], C2×Q8 [×2], C4○D4 [×8], Dic5 [×4], C20 [×4], C20 [×5], D10 [×4], D10 [×10], C2×C10, C2×C4⋊C4 [×2], C4×D4 [×2], C4⋊D4 [×2], C22⋊Q8 [×2], C22.D4 [×4], C4⋊Q8, C2×C4○D4 [×2], C4×D5 [×16], D20 [×4], D20 [×10], C2×Dic5 [×4], C2×C20 [×3], C2×C20 [×4], C5×Q8 [×4], C22×D5 [×4], D46D4, C10.D4 [×4], C4⋊Dic5 [×2], D10⋊C4 [×8], C4×C20, C5×C4⋊C4 [×4], C2×C4×D5 [×8], C2×D20 [×2], C2×D20 [×4], Q82D5 [×8], Q8×C10 [×2], C4×D20 [×2], D5×C4⋊C4 [×2], D10.13D4 [×4], C4⋊D20 [×2], D103Q8 [×2], C5×C4⋊Q8, C2×Q82D5 [×2], D2012D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C4○D4 [×2], C24, D10 [×7], C22×D4, C2×C4○D4, 2- 1+4, C22×D5 [×7], D46D4, D4×D5 [×2], Q82D5 [×2], C23×D5, C2×D4×D5, C2×Q82D5, Q8.10D10, D2012D4

Smallest permutation representation of D2012D4
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 57)(2 56)(3 55)(4 54)(5 53)(6 52)(7 51)(8 50)(9 49)(10 48)(11 47)(12 46)(13 45)(14 44)(15 43)(16 42)(17 41)(18 60)(19 59)(20 58)(21 103)(22 102)(23 101)(24 120)(25 119)(26 118)(27 117)(28 116)(29 115)(30 114)(31 113)(32 112)(33 111)(34 110)(35 109)(36 108)(37 107)(38 106)(39 105)(40 104)(61 159)(62 158)(63 157)(64 156)(65 155)(66 154)(67 153)(68 152)(69 151)(70 150)(71 149)(72 148)(73 147)(74 146)(75 145)(76 144)(77 143)(78 142)(79 141)(80 160)(81 139)(82 138)(83 137)(84 136)(85 135)(86 134)(87 133)(88 132)(89 131)(90 130)(91 129)(92 128)(93 127)(94 126)(95 125)(96 124)(97 123)(98 122)(99 121)(100 140)
(1 38 156 140)(2 39 157 121)(3 40 158 122)(4 21 159 123)(5 22 160 124)(6 23 141 125)(7 24 142 126)(8 25 143 127)(9 26 144 128)(10 27 145 129)(11 28 146 130)(12 29 147 131)(13 30 148 132)(14 31 149 133)(15 32 150 134)(16 33 151 135)(17 34 152 136)(18 35 153 137)(19 36 154 138)(20 37 155 139)(41 110 68 84)(42 111 69 85)(43 112 70 86)(44 113 71 87)(45 114 72 88)(46 115 73 89)(47 116 74 90)(48 117 75 91)(49 118 76 92)(50 119 77 93)(51 120 78 94)(52 101 79 95)(53 102 80 96)(54 103 61 97)(55 104 62 98)(56 105 63 99)(57 106 64 100)(58 107 65 81)(59 108 66 82)(60 109 67 83)
(1 156)(2 145)(3 154)(4 143)(5 152)(6 141)(7 150)(8 159)(9 148)(10 157)(11 146)(12 155)(13 144)(14 153)(15 142)(16 151)(17 160)(18 149)(19 158)(20 147)(21 25)(22 34)(24 32)(26 30)(27 39)(29 37)(31 35)(36 40)(41 62)(42 71)(43 80)(44 69)(45 78)(46 67)(47 76)(48 65)(49 74)(50 63)(51 72)(52 61)(53 70)(54 79)(55 68)(56 77)(57 66)(58 75)(59 64)(60 73)(81 91)(82 100)(83 89)(84 98)(85 87)(86 96)(88 94)(90 92)(93 99)(95 97)(101 103)(102 112)(104 110)(105 119)(106 108)(107 117)(109 115)(111 113)(114 120)(116 118)(121 129)(122 138)(123 127)(124 136)(126 134)(128 132)(131 139)(133 137)```

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

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

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

53 conjugacy classes

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

53 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + - + + image C1 C2 C2 C2 C2 C2 C2 C2 D4 D5 C4○D4 D10 D10 D10 2- 1+4 D4×D5 Q8⋊2D5 Q8.10D10 kernel D20⋊12D4 C4×D20 D5×C4⋊C4 D10.13D4 C4⋊D20 D10⋊3Q8 C5×C4⋊Q8 C2×Q8⋊2D5 D20 C4⋊Q8 C20 C42 C4⋊C4 C2×Q8 C10 C4 C4 C2 # reps 1 2 2 4 2 2 1 2 4 2 4 2 8 4 1 4 4 4

Matrix representation of D2012D4 in GL6(𝔽41)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 40 0 0 0 0 8 34 0 0 0 0 0 0 32 0 0 0 0 0 2 9
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 8 40 0 0 0 0 0 0 32 1 0 0 0 0 2 9
,
 1 39 0 0 0 0 1 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 40 0 0 0 0 0 40 1 0 0 0 0 0 0 34 1 0 0 0 0 34 7 0 0 0 0 0 0 1 0 0 0 0 0 18 40

`G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,8,0,0,0,0,40,34,0,0,0,0,0,0,32,2,0,0,0,0,0,9],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,8,0,0,0,0,0,40,0,0,0,0,0,0,32,2,0,0,0,0,1,9],[1,1,0,0,0,0,39,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,40,0,0,0,0,0,1,0,0,0,0,0,0,34,34,0,0,0,0,1,7,0,0,0,0,0,0,1,18,0,0,0,0,0,40] >;`

D2012D4 in GAP, Magma, Sage, TeX

`D_{20}\rtimes_{12}D_4`
`% in TeX`

`G:=Group("D20:12D4");`
`// GroupNames label`

`G:=SmallGroup(320,1398);`
`// by ID`

`G=gap.SmallGroup(320,1398);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,219,268,1571,297,136,12550]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^20=b^2=c^4=d^2=1,b*a*b=a^-1,a*c=c*a,d*a*d=a^9,b*c=c*b,d*b*d=a^18*b,d*c*d=c^-1>;`
`// generators/relations`

׿
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