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## G = D20⋊10D4order 320 = 26·5

### 3rd semidirect product of D20 and D4 acting via D4/C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — D20⋊10D4
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C23×D5 — C2×D4×D5 — D20⋊10D4
 Lower central C5 — C2×C10 — D20⋊10D4
 Upper central C1 — C22 — C4.4D4

Generators and relations for D2010D4
G = < a,b,c,d | a20=b2=c4=d2=1, bab=a-1, ac=ca, dad=a9, cbc-1=a10b, dbd=a18b, dcd=c-1 >

Subgroups: 1478 in 334 conjugacy classes, 105 normal (43 characteristic)
C1, C2 [×3], C2 [×9], C4 [×2], C4 [×8], C22, C22 [×29], C5, C2×C4 [×3], C2×C4 [×2], C2×C4 [×14], D4 [×18], Q8 [×2], C23 [×2], C23 [×14], D5 [×7], C10 [×3], C10 [×2], C42, C22⋊C4 [×4], C22⋊C4 [×8], C4⋊C4 [×4], C22×C4 [×6], C2×D4, C2×D4 [×12], C2×Q8, C4○D4 [×4], C24 [×2], Dic5 [×4], C20 [×2], C20 [×4], D10 [×6], D10 [×17], C2×C10, C2×C10 [×6], C2×C22⋊C4 [×2], C4×D4 [×2], C22≀C2 [×2], C4⋊D4 [×3], C22⋊Q8, C22.D4 [×2], C4.4D4, C22×D4, C2×C4○D4, C4×D5 [×10], D20 [×4], D20 [×6], C2×Dic5 [×2], C2×Dic5 [×2], C5⋊D4 [×6], C2×C20 [×3], C2×C20 [×2], C5×D4 [×2], C5×Q8 [×2], C22×D5 [×2], C22×D5 [×2], C22×D5 [×10], C22×C10 [×2], D45D4, C10.D4 [×2], C4⋊Dic5 [×2], D10⋊C4 [×2], D10⋊C4 [×4], C23.D5 [×2], C4×C20, C5×C22⋊C4 [×4], C2×C4×D5 [×2], C2×C4×D5 [×4], C2×D20 [×2], C2×D20 [×2], D4×D5 [×4], Q82D5 [×4], C2×C5⋊D4 [×4], D4×C10, Q8×C10, C23×D5 [×2], C4×D20 [×2], D5×C22⋊C4 [×2], C22⋊D20 [×2], D10.12D4 [×2], D10⋊D4 [×2], C202D4, D103Q8, C5×C4.4D4, C2×D4×D5, C2×Q82D5, D2010D4
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, D5×C4○D4, D48D10, D2010D4

Smallest permutation representation of D2010D4
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 52)(42 51)(43 50)(44 49)(45 48)(46 47)(53 60)(54 59)(55 58)(56 57)(61 64)(62 63)(65 80)(66 79)(67 78)(68 77)(69 76)(70 75)(71 74)(72 73)
(1 68 39 52)(2 69 40 53)(3 70 21 54)(4 71 22 55)(5 72 23 56)(6 73 24 57)(7 74 25 58)(8 75 26 59)(9 76 27 60)(10 77 28 41)(11 78 29 42)(12 79 30 43)(13 80 31 44)(14 61 32 45)(15 62 33 46)(16 63 34 47)(17 64 35 48)(18 65 36 49)(19 66 37 50)(20 67 38 51)
(1 78)(2 67)(3 76)(4 65)(5 74)(6 63)(7 72)(8 61)(9 70)(10 79)(11 68)(12 77)(13 66)(14 75)(15 64)(16 73)(17 62)(18 71)(19 80)(20 69)(21 60)(22 49)(23 58)(24 47)(25 56)(26 45)(27 54)(28 43)(29 52)(30 41)(31 50)(32 59)(33 48)(34 57)(35 46)(36 55)(37 44)(38 53)(39 42)(40 51)```

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

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

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

53 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F ··· 2K 2L 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 5A 5B 10A ··· 10F 10G 10H 10I 10J 20A ··· 20L 20M 20N 20O 20P 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 20 ··· 20 20 20 20 20 size 1 1 1 1 4 4 10 ··· 10 20 2 2 2 2 4 4 4 10 10 20 20 20 2 2 2 ··· 2 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 D5 C4○D4 D10 D10 D10 D10 2+ 1+4 D4×D5 D5×C4○D4 D4⋊8D10 kernel D20⋊10D4 C4×D20 D5×C22⋊C4 C22⋊D20 D10.12D4 D10⋊D4 C20⋊2D4 D10⋊3Q8 C5×C4.4D4 C2×D4×D5 C2×Q8⋊2D5 D20 C4.4D4 D10 C42 C22⋊C4 C2×D4 C2×Q8 C10 C4 C2 C2 # reps 1 2 2 2 2 2 1 1 1 1 1 4 2 4 2 8 2 2 1 4 4 4

Matrix representation of D2010D4 in GL6(𝔽41)

 7 1 0 0 0 0 33 40 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 16 40 0 0 0 0 11 25
,
 40 40 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 16 40 0 0 0 0 9 25
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 1 39 0 0 0 0 1 40 0 0 0 0 0 0 20 9 0 0 0 0 24 21
,
 7 6 0 0 0 0 33 34 0 0 0 0 0 0 40 2 0 0 0 0 0 1 0 0 0 0 0 0 21 32 0 0 0 0 17 20

`G:=sub<GL(6,GF(41))| [7,33,0,0,0,0,1,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,11,0,0,0,0,40,25],[40,0,0,0,0,0,40,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,9,0,0,0,0,40,25],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,1,0,0,0,0,39,40,0,0,0,0,0,0,20,24,0,0,0,0,9,21],[7,33,0,0,0,0,6,34,0,0,0,0,0,0,40,0,0,0,0,0,2,1,0,0,0,0,0,0,21,17,0,0,0,0,32,20] >;`

D2010D4 in GAP, Magma, Sage, TeX

`D_{20}\rtimes_{10}D_4`
`% in TeX`

`G:=Group("D20:10D4");`
`// GroupNames label`

`G:=SmallGroup(320,1348);`
`// by ID`

`G=gap.SmallGroup(320,1348);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,219,1571,570,297,192,12550]);`
`// 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,c*b*c^-1=a^10*b,d*b*d=a^18*b,d*c*d=c^-1>;`
`// generators/relations`

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