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## G = D20⋊19D4order 320 = 26·5

### 7th semidirect product of D20 and D4 acting via D4/C22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — D20⋊19D4
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C23×D5 — C2×D4×D5 — D20⋊19D4
 Lower central C5 — C2×C10 — D20⋊19D4
 Upper central C1 — C22 — C4⋊D4

Generators and relations for D2019D4
G = < a,b,c,d | a20=b2=c4=d2=1, bab=a-1, cac-1=dad=a11, cbc-1=dbd=a10b, dcd=c-1 >

Subgroups: 2038 in 428 conjugacy classes, 115 normal (43 characteristic)
C1, C2 [×3], C2 [×12], C4 [×2], C4 [×7], C22, C22 [×2], C22 [×42], C5, C2×C4 [×2], C2×C4 [×2], C2×C4 [×11], D4 [×34], C23, C23 [×2], C23 [×25], D5 [×8], C10 [×3], C10 [×4], C42, C22⋊C4 [×2], C22⋊C4 [×6], C4⋊C4, C4⋊C4, C22×C4, C22×C4 [×3], C2×D4, C2×D4 [×2], C2×D4 [×29], C24 [×4], Dic5 [×2], Dic5 [×2], C20 [×2], C20 [×3], D10 [×6], D10 [×28], C2×C10, C2×C10 [×2], C2×C10 [×8], C4×D4 [×2], C22≀C2 [×4], C4⋊D4, C4⋊D4 [×3], C41D4, C22×D4 [×4], C4×D5 [×6], D20 [×4], D20 [×10], C2×Dic5 [×3], C5⋊D4 [×4], C5⋊D4 [×10], C2×C20 [×2], C2×C20 [×2], C2×C20 [×2], C5×D4 [×6], C22×D5, C22×D5 [×4], C22×D5 [×20], C22×C10, C22×C10 [×2], D42, C4×Dic5, C10.D4, D10⋊C4, D10⋊C4 [×4], C23.D5, C5×C22⋊C4 [×2], C5×C4⋊C4, C2×C4×D5, C2×C4×D5 [×2], C2×D20 [×2], C2×D20 [×4], C2×D20 [×4], D4×D5 [×12], C2×C5⋊D4, C2×C5⋊D4 [×6], C22×C20, D4×C10, D4×C10 [×2], C23×D5 [×4], C22⋊D20 [×2], D10⋊D4 [×2], D208C4, C4⋊D20, C4×C5⋊D4, C23⋊D10 [×2], C20⋊D4, C5×C4⋊D4, C22×D20, C2×D4×D5, C2×D4×D5 [×2], D2019D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], D5, C2×D4 [×12], C24, D10 [×7], C22×D4 [×2], 2+ 1+4, C22×D5 [×7], D42, D4×D5 [×4], C23×D5, C2×D4×D5 [×2], D48D10, D2019D4

Smallest permutation representation of D2019D4
On 80 points
Generators in S80
```(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)
(1 20)(2 19)(3 18)(4 17)(5 16)(6 15)(7 14)(8 13)(9 12)(10 11)(21 32)(22 31)(23 30)(24 29)(25 28)(26 27)(33 40)(34 39)(35 38)(36 37)(41 50)(42 49)(43 48)(44 47)(45 46)(51 60)(52 59)(53 58)(54 57)(55 56)(61 72)(62 71)(63 70)(64 69)(65 68)(66 67)(73 80)(74 79)(75 78)(76 77)
(1 46 67 37)(2 57 68 28)(3 48 69 39)(4 59 70 30)(5 50 71 21)(6 41 72 32)(7 52 73 23)(8 43 74 34)(9 54 75 25)(10 45 76 36)(11 56 77 27)(12 47 78 38)(13 58 79 29)(14 49 80 40)(15 60 61 31)(16 51 62 22)(17 42 63 33)(18 53 64 24)(19 44 65 35)(20 55 66 26)
(1 77)(2 68)(3 79)(4 70)(5 61)(6 72)(7 63)(8 74)(9 65)(10 76)(11 67)(12 78)(13 69)(14 80)(15 71)(16 62)(17 73)(18 64)(19 75)(20 66)(21 31)(23 33)(25 35)(27 37)(29 39)(42 52)(44 54)(46 56)(48 58)(50 60)```

`G:=sub<Sym(80)| (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), (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,11)(21,32)(22,31)(23,30)(24,29)(25,28)(26,27)(33,40)(34,39)(35,38)(36,37)(41,50)(42,49)(43,48)(44,47)(45,46)(51,60)(52,59)(53,58)(54,57)(55,56)(61,72)(62,71)(63,70)(64,69)(65,68)(66,67)(73,80)(74,79)(75,78)(76,77), (1,46,67,37)(2,57,68,28)(3,48,69,39)(4,59,70,30)(5,50,71,21)(6,41,72,32)(7,52,73,23)(8,43,74,34)(9,54,75,25)(10,45,76,36)(11,56,77,27)(12,47,78,38)(13,58,79,29)(14,49,80,40)(15,60,61,31)(16,51,62,22)(17,42,63,33)(18,53,64,24)(19,44,65,35)(20,55,66,26), (1,77)(2,68)(3,79)(4,70)(5,61)(6,72)(7,63)(8,74)(9,65)(10,76)(11,67)(12,78)(13,69)(14,80)(15,71)(16,62)(17,73)(18,64)(19,75)(20,66)(21,31)(23,33)(25,35)(27,37)(29,39)(42,52)(44,54)(46,56)(48,58)(50,60)>;`

`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), (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,11)(21,32)(22,31)(23,30)(24,29)(25,28)(26,27)(33,40)(34,39)(35,38)(36,37)(41,50)(42,49)(43,48)(44,47)(45,46)(51,60)(52,59)(53,58)(54,57)(55,56)(61,72)(62,71)(63,70)(64,69)(65,68)(66,67)(73,80)(74,79)(75,78)(76,77), (1,46,67,37)(2,57,68,28)(3,48,69,39)(4,59,70,30)(5,50,71,21)(6,41,72,32)(7,52,73,23)(8,43,74,34)(9,54,75,25)(10,45,76,36)(11,56,77,27)(12,47,78,38)(13,58,79,29)(14,49,80,40)(15,60,61,31)(16,51,62,22)(17,42,63,33)(18,53,64,24)(19,44,65,35)(20,55,66,26), (1,77)(2,68)(3,79)(4,70)(5,61)(6,72)(7,63)(8,74)(9,65)(10,76)(11,67)(12,78)(13,69)(14,80)(15,71)(16,62)(17,73)(18,64)(19,75)(20,66)(21,31)(23,33)(25,35)(27,37)(29,39)(42,52)(44,54)(46,56)(48,58)(50,60) );`

`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)], [(1,20),(2,19),(3,18),(4,17),(5,16),(6,15),(7,14),(8,13),(9,12),(10,11),(21,32),(22,31),(23,30),(24,29),(25,28),(26,27),(33,40),(34,39),(35,38),(36,37),(41,50),(42,49),(43,48),(44,47),(45,46),(51,60),(52,59),(53,58),(54,57),(55,56),(61,72),(62,71),(63,70),(64,69),(65,68),(66,67),(73,80),(74,79),(75,78),(76,77)], [(1,46,67,37),(2,57,68,28),(3,48,69,39),(4,59,70,30),(5,50,71,21),(6,41,72,32),(7,52,73,23),(8,43,74,34),(9,54,75,25),(10,45,76,36),(11,56,77,27),(12,47,78,38),(13,58,79,29),(14,49,80,40),(15,60,61,31),(16,51,62,22),(17,42,63,33),(18,53,64,24),(19,44,65,35),(20,55,66,26)], [(1,77),(2,68),(3,79),(4,70),(5,61),(6,72),(7,63),(8,74),(9,65),(10,76),(11,67),(12,78),(13,69),(14,80),(15,71),(16,62),(17,73),(18,64),(19,75),(20,66),(21,31),(23,33),(25,35),(27,37),(29,39),(42,52),(44,54),(46,56),(48,58),(50,60)])`

53 conjugacy classes

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

53 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 D4 D4 D5 D10 D10 D10 D10 2+ 1+4 D4×D5 D4×D5 D4⋊8D10 kernel D20⋊19D4 C22⋊D20 D10⋊D4 D20⋊8C4 C4⋊D20 C4×C5⋊D4 C23⋊D10 C20⋊D4 C5×C4⋊D4 C22×D20 C2×D4×D5 D20 C5⋊D4 C4⋊D4 C22⋊C4 C4⋊C4 C22×C4 C2×D4 C10 C4 C22 C2 # reps 1 2 2 1 1 1 2 1 1 1 3 4 4 2 4 2 2 6 1 4 4 4

Matrix representation of D2019D4 in GL6(𝔽41)

 40 0 0 0 0 0 0 40 0 0 0 0 0 0 0 1 0 0 0 0 40 6 0 0 0 0 0 0 0 1 0 0 0 0 40 0
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 0 1 0 0 0 0 40 0 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 40 0 0 0 0 0 0 1

`G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,40,0,0,0,0,1,6,0,0,0,0,0,0,0,40,0,0,0,0,1,0],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[0,40,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,1] >;`

D2019D4 in GAP, Magma, Sage, TeX

`D_{20}\rtimes_{19}D_4`
`% in TeX`

`G:=Group("D20:19D4");`
`// GroupNames label`

`G:=SmallGroup(320,1281);`
`// by ID`

`G=gap.SmallGroup(320,1281);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,219,1571,297,192,12550]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^20=b^2=c^4=d^2=1,b*a*b=a^-1,c*a*c^-1=d*a*d=a^11,c*b*c^-1=d*b*d=a^10*b,d*c*d=c^-1>;`
`// generators/relations`

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