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## G = D20⋊21D4order 320 = 26·5

### 9th semidirect product of D20 and D4 acting via D4/C22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — D20⋊21D4
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C23×D5 — D5×C22⋊C4 — D20⋊21D4
 Lower central C5 — C2×C10 — D20⋊21D4
 Upper central C1 — C22 — C22⋊Q8

Generators and relations for D2021D4
G = < a,b,c,d | a20=b2=c4=d2=1, bab=a-1, cac-1=dad=a9, cbc-1=dbd=a18b, dcd=c-1 >

Subgroups: 1510 in 334 conjugacy classes, 107 normal (43 characteristic)
C1, C2 [×3], C2 [×9], C4 [×2], C4 [×8], C22, C22 [×2], C22 [×27], C5, C2×C4 [×2], C2×C4 [×4], C2×C4 [×13], D4 [×18], Q8 [×2], C23, C23 [×15], D5 [×7], C10 [×3], C10 [×2], C42, C22⋊C4 [×2], C22⋊C4 [×10], C4⋊C4, C4⋊C4 [×2], C4⋊C4, C22×C4, C22×C4 [×5], C2×D4 [×13], C2×Q8, C4○D4 [×4], C24 [×2], Dic5 [×3], C20 [×2], C20 [×5], D10 [×4], D10 [×21], C2×C10, C2×C10 [×2], C2×C10 [×2], 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 [×8], D20 [×4], D20 [×12], C2×Dic5 [×3], C5⋊D4 [×2], C2×C20 [×2], C2×C20 [×4], C2×C20 [×2], C5×Q8 [×2], C22×D5, C22×D5 [×4], C22×D5 [×10], C22×C10, D45D4, C4×Dic5, C10.D4, D10⋊C4, D10⋊C4 [×8], C23.D5, C5×C22⋊C4 [×2], C5×C4⋊C4, C5×C4⋊C4 [×2], C2×C4×D5, C2×C4×D5 [×4], C2×D20 [×2], C2×D20 [×6], C2×D20 [×4], Q82D5 [×4], C2×C5⋊D4, C22×C20, Q8×C10, C23×D5 [×2], D5×C22⋊C4 [×2], C22⋊D20 [×2], D208C4, D10.13D4 [×2], C4⋊D20, C4⋊D20 [×2], C4×C5⋊D4, C20.23D4, C5×C22⋊Q8, C22×D20, C2×Q82D5, D2021D4
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], Q82D5 [×2], C23×D5, C2×D4×D5, C2×Q82D5, D48D10, D2021D4

Smallest permutation representation of D2021D4
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 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 20)(17 19)(22 40)(23 39)(24 38)(25 37)(26 36)(27 35)(28 34)(29 33)(30 32)(41 57)(42 56)(43 55)(44 54)(45 53)(46 52)(47 51)(48 50)(58 60)(62 80)(63 79)(64 78)(65 77)(66 76)(67 75)(68 74)(69 73)(70 72)
(1 39 52 79)(2 28 53 68)(3 37 54 77)(4 26 55 66)(5 35 56 75)(6 24 57 64)(7 33 58 73)(8 22 59 62)(9 31 60 71)(10 40 41 80)(11 29 42 69)(12 38 43 78)(13 27 44 67)(14 36 45 76)(15 25 46 65)(16 34 47 74)(17 23 48 63)(18 32 49 72)(19 21 50 61)(20 30 51 70)
(1 69)(2 78)(3 67)(4 76)(5 65)(6 74)(7 63)(8 72)(9 61)(10 70)(11 79)(12 68)(13 77)(14 66)(15 75)(16 64)(17 73)(18 62)(19 71)(20 80)(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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,20)(17,19)(22,40)(23,39)(24,38)(25,37)(26,36)(27,35)(28,34)(29,33)(30,32)(41,57)(42,56)(43,55)(44,54)(45,53)(46,52)(47,51)(48,50)(58,60)(62,80)(63,79)(64,78)(65,77)(66,76)(67,75)(68,74)(69,73)(70,72), (1,39,52,79)(2,28,53,68)(3,37,54,77)(4,26,55,66)(5,35,56,75)(6,24,57,64)(7,33,58,73)(8,22,59,62)(9,31,60,71)(10,40,41,80)(11,29,42,69)(12,38,43,78)(13,27,44,67)(14,36,45,76)(15,25,46,65)(16,34,47,74)(17,23,48,63)(18,32,49,72)(19,21,50,61)(20,30,51,70), (1,69)(2,78)(3,67)(4,76)(5,65)(6,74)(7,63)(8,72)(9,61)(10,70)(11,79)(12,68)(13,77)(14,66)(15,75)(16,64)(17,73)(18,62)(19,71)(20,80)(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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,20)(17,19)(22,40)(23,39)(24,38)(25,37)(26,36)(27,35)(28,34)(29,33)(30,32)(41,57)(42,56)(43,55)(44,54)(45,53)(46,52)(47,51)(48,50)(58,60)(62,80)(63,79)(64,78)(65,77)(66,76)(67,75)(68,74)(69,73)(70,72), (1,39,52,79)(2,28,53,68)(3,37,54,77)(4,26,55,66)(5,35,56,75)(6,24,57,64)(7,33,58,73)(8,22,59,62)(9,31,60,71)(10,40,41,80)(11,29,42,69)(12,38,43,78)(13,27,44,67)(14,36,45,76)(15,25,46,65)(16,34,47,74)(17,23,48,63)(18,32,49,72)(19,21,50,61)(20,30,51,70), (1,69)(2,78)(3,67)(4,76)(5,65)(6,74)(7,63)(8,72)(9,61)(10,70)(11,79)(12,68)(13,77)(14,66)(15,75)(16,64)(17,73)(18,62)(19,71)(20,80)(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)], [(1,15),(2,14),(3,13),(4,12),(5,11),(6,10),(7,9),(16,20),(17,19),(22,40),(23,39),(24,38),(25,37),(26,36),(27,35),(28,34),(29,33),(30,32),(41,57),(42,56),(43,55),(44,54),(45,53),(46,52),(47,51),(48,50),(58,60),(62,80),(63,79),(64,78),(65,77),(66,76),(67,75),(68,74),(69,73),(70,72)], [(1,39,52,79),(2,28,53,68),(3,37,54,77),(4,26,55,66),(5,35,56,75),(6,24,57,64),(7,33,58,73),(8,22,59,62),(9,31,60,71),(10,40,41,80),(11,29,42,69),(12,38,43,78),(13,27,44,67),(14,36,45,76),(15,25,46,65),(16,34,47,74),(17,23,48,63),(18,32,49,72),(19,21,50,61),(20,30,51,70)], [(1,69),(2,78),(3,67),(4,76),(5,65),(6,74),(7,63),(8,72),(9,61),(10,70),(11,79),(12,68),(13,77),(14,66),(15,75),(16,64),(17,73),(18,62),(19,71),(20,80),(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 2G 2H 2I 2J 2K 2L 4A 4B 4C ··· 4G 4H 4I 4J 4K 4L 5A 5B 10A ··· 10F 10G 10H 10I 10J 20A ··· 20H 20I ··· 20P order 1 2 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 20 ··· 20 20 ··· 20 size 1 1 1 1 2 2 10 10 10 10 20 20 20 2 2 4 ··· 4 10 10 10 10 20 2 2 2 ··· 2 4 4 4 4 4 ··· 4 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 Q8⋊2D5 D4⋊8D10 kernel D20⋊21D4 D5×C22⋊C4 C22⋊D20 D20⋊8C4 D10.13D4 C4⋊D20 C4×C5⋊D4 C20.23D4 C5×C22⋊Q8 C22×D20 C2×Q8⋊2D5 D20 C22⋊Q8 C2×C10 C22⋊C4 C4⋊C4 C22×C4 C2×Q8 C10 C4 C22 C2 # reps 1 2 2 1 2 3 1 1 1 1 1 4 2 4 4 6 2 2 1 4 4 4

Matrix representation of D2021D4 in GL6(𝔽41)

 0 1 0 0 0 0 40 6 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 36 0 0 0 0 25 1
,
 0 1 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 40 0 0 0 0 0 25 1
,
 40 0 0 0 0 0 35 1 0 0 0 0 0 0 0 1 0 0 0 0 40 0 0 0 0 0 0 0 32 37 0 0 0 0 20 9
,
 40 0 0 0 0 0 35 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 9 4 0 0 0 0 21 32

`G:=sub<GL(6,GF(41))| [0,40,0,0,0,0,1,6,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,25,0,0,0,0,36,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,25,0,0,0,0,0,1],[40,35,0,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,1,0,0,0,0,0,0,0,32,20,0,0,0,0,37,9],[40,35,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,9,21,0,0,0,0,4,32] >;`

D2021D4 in GAP, Magma, Sage, TeX

`D_{20}\rtimes_{21}D_4`
`% in TeX`

`G:=Group("D20:21D4");`
`// GroupNames label`

`G:=SmallGroup(320,1302);`
`// by ID`

`G=gap.SmallGroup(320,1302);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,219,184,1571,297,192,12550]);`
`// 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^9,c*b*c^-1=d*b*d=a^18*b,d*c*d=c^-1>;`
`// generators/relations`

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