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## G = D20⋊20D4order 320 = 26·5

### 8th semidirect product of D20 and D4 acting via D4/C22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — D20⋊20D4
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C23×D5 — C2×D4×D5 — D20⋊20D4
 Lower central C5 — C2×C10 — D20⋊20D4
 Upper central C1 — C22 — C4⋊D4

Generators and relations for D2020D4
G = < a,b,c,d | a20=b2=c4=d2=1, bab=a-1, cac-1=dad=a11, bc=cb, dbd=a10b, dcd=c-1 >

Subgroups: 1414 in 334 conjugacy classes, 105 normal (43 characteristic)
C1, C2 [×3], C2 [×9], C4 [×2], C4 [×8], C22, C22 [×29], C5, C2×C4 [×2], C2×C4 [×2], C2×C4 [×15], D4 [×18], Q8 [×2], C23, C23 [×2], C23 [×13], D5 [×6], C10 [×3], C10 [×3], C42, C22⋊C4 [×2], C22⋊C4 [×10], C4⋊C4, C4⋊C4 [×3], C22×C4, C22×C4 [×5], C2×D4, C2×D4 [×2], C2×D4 [×10], C2×Q8, C4○D4 [×4], C24 [×2], Dic5 [×5], C20 [×2], C20 [×3], D10 [×6], D10 [×14], C2×C10, C2×C10 [×9], C2×C22⋊C4 [×2], C4×D4 [×2], C22≀C2 [×2], C4⋊D4, C4⋊D4 [×2], C22⋊Q8, C22.D4 [×2], C4.4D4, C22×D4, C2×C4○D4, Dic10 [×2], C4×D5 [×8], D20 [×4], C2×Dic5 [×3], C2×Dic5 [×2], C5⋊D4 [×10], C2×C20 [×2], C2×C20 [×2], C2×C20 [×2], C5×D4 [×4], C22×D5, C22×D5 [×2], C22×D5 [×10], C22×C10, C22×C10 [×2], D45D4, C4×Dic5, C10.D4, C4⋊Dic5 [×2], D10⋊C4, D10⋊C4 [×4], C23.D5, C23.D5 [×4], C5×C22⋊C4 [×2], C5×C4⋊C4, C2×Dic10, C2×C4×D5, C2×C4×D5 [×4], C2×D20, C4○D20 [×4], D4×D5 [×4], C2×C5⋊D4, C2×C5⋊D4 [×4], C22×C20, D4×C10, D4×C10 [×2], C23×D5 [×2], D5×C22⋊C4 [×2], D10.12D4 [×2], D208C4, D102Q8, C4×C5⋊D4, C20.17D4, C23⋊D10 [×2], C202D4 [×2], C5×C4⋊D4, C2×C4○D20, C2×D4×D5, D2020D4
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], D45D4, D4×D5 [×2], C23×D5, C2×D4×D5, D46D10, D5×C4○D4, D2020D4

Smallest permutation representation of D2020D4
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 36)(22 35)(23 34)(24 33)(25 32)(26 31)(27 30)(28 29)(37 40)(38 39)(41 58)(42 57)(43 56)(44 55)(45 54)(46 53)(47 52)(48 51)(49 50)(59 60)(61 70)(62 69)(63 68)(64 67)(65 66)(71 80)(72 79)(73 78)(74 77)(75 76)
(1 71 50 34)(2 62 51 25)(3 73 52 36)(4 64 53 27)(5 75 54 38)(6 66 55 29)(7 77 56 40)(8 68 57 31)(9 79 58 22)(10 70 59 33)(11 61 60 24)(12 72 41 35)(13 63 42 26)(14 74 43 37)(15 65 44 28)(16 76 45 39)(17 67 46 30)(18 78 47 21)(19 69 48 32)(20 80 49 23)
(1 50)(2 41)(3 52)(4 43)(5 54)(6 45)(7 56)(8 47)(9 58)(10 49)(11 60)(12 51)(13 42)(14 53)(15 44)(16 55)(17 46)(18 57)(19 48)(20 59)(21 31)(23 33)(25 35)(27 37)(29 39)(62 72)(64 74)(66 76)(68 78)(70 80)```

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

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

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

53 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G ··· 2L 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 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 4 4 4 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 4 4 4 10 ··· 10 2 2 2 2 4 4 10 10 20 20 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 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 C2 D4 D5 C4○D4 D10 D10 D10 D10 2+ 1+4 D4×D5 D4⋊6D10 D5×C4○D4 kernel D20⋊20D4 D5×C22⋊C4 D10.12D4 D20⋊8C4 D10⋊2Q8 C4×C5⋊D4 C20.17D4 C23⋊D10 C20⋊2D4 C5×C4⋊D4 C2×C4○D20 C2×D4×D5 D20 C4⋊D4 D10 C22⋊C4 C4⋊C4 C22×C4 C2×D4 C10 C4 C2 C2 # reps 1 2 2 1 1 1 1 2 2 1 1 1 4 2 4 4 2 2 6 1 4 4 4

Matrix representation of D2020D4 in GL6(𝔽41)

 0 32 0 0 0 0 32 0 0 0 0 0 0 0 7 1 0 0 0 0 33 40 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 32 0 0 0 0 9 0 0 0 0 0 0 0 40 40 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 1 0 0 0 0 40 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 40 0
,
 40 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 0 40 0 0 0 0 0 0 1

`G:=sub<GL(6,GF(41))| [0,32,0,0,0,0,32,0,0,0,0,0,0,0,7,33,0,0,0,0,1,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,9,0,0,0,0,32,0,0,0,0,0,0,0,40,0,0,0,0,0,40,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,40,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,1,0],[40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,1] >;`

D2020D4 in GAP, Magma, Sage, TeX

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

`G:=Group("D20:20D4");`
`// GroupNames label`

`G:=SmallGroup(320,1284);`
`// by ID`

`G=gap.SmallGroup(320,1284);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,219,1571,570,297,12550]);`
`// 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,b*c=c*b,d*b*d=a^10*b,d*c*d=c^-1>;`
`// generators/relations`

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