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## G = D20⋊1D4order 320 = 26·5

### 1st semidirect product of D20 and D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — D20⋊1D4
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C4○D20 — D4⋊6D10 — D20⋊1D4
 Lower central C5 — C10 — C2×C20 — D20⋊1D4
 Upper central C1 — C2 — C2×C4 — C4.D4

Generators and relations for D201D4
G = < a,b,c,d | a20=b2=c4=d2=1, bab=dad=a-1, cac-1=a9, cbc-1=a13b, dbd=a3b, dcd=c-1 >

Subgroups: 942 in 168 conjugacy classes, 39 normal (17 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, Q8, C23, C23, D5, C10, C10, C42, M4(2), D8, SD16, C2×D4, C2×D4, C4○D4, Dic5, C20, D10, C2×C10, C2×C10, C4.D4, C4≀C2, C41D4, C8⋊C22, 2+ 1+4, C40, Dic10, C4×D5, D20, D20, C2×Dic5, C5⋊D4, C2×C20, C5×D4, C22×D5, C22×C10, D44D4, C40⋊C2, D40, C4×Dic5, C5×M4(2), C2×D20, C4○D20, D4×D5, D42D5, C2×C5⋊D4, D4×C10, D207C4, C5×C4.D4, C8⋊D10, C20⋊D4, D46D10, D201D4
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, D10, C22≀C2, D20, C22×D5, D44D4, C2×D20, D4×D5, C22⋊D20, D201D4

Smallest permutation representation of D201D4
On 40 points
Generators in S40
```(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)
(1 36)(2 35)(3 34)(4 33)(5 32)(6 31)(7 30)(8 29)(9 28)(10 27)(11 26)(12 25)(13 24)(14 23)(15 22)(16 21)(17 40)(18 39)(19 38)(20 37)
(1 6 11 16)(2 15 12 5)(3 4 13 14)(7 20 17 10)(8 9 18 19)(21 33)(23 31)(24 40)(25 29)(26 38)(28 36)(30 34)(35 39)
(2 20)(3 19)(4 18)(5 17)(6 16)(7 15)(8 14)(9 13)(10 12)(21 28)(22 27)(23 26)(24 25)(29 40)(30 39)(31 38)(32 37)(33 36)(34 35)```

`G:=sub<Sym(40)| (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), (1,36)(2,35)(3,34)(4,33)(5,32)(6,31)(7,30)(8,29)(9,28)(10,27)(11,26)(12,25)(13,24)(14,23)(15,22)(16,21)(17,40)(18,39)(19,38)(20,37), (1,6,11,16)(2,15,12,5)(3,4,13,14)(7,20,17,10)(8,9,18,19)(21,33)(23,31)(24,40)(25,29)(26,38)(28,36)(30,34)(35,39), (2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,12)(21,28)(22,27)(23,26)(24,25)(29,40)(30,39)(31,38)(32,37)(33,36)(34,35)>;`

`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), (1,36)(2,35)(3,34)(4,33)(5,32)(6,31)(7,30)(8,29)(9,28)(10,27)(11,26)(12,25)(13,24)(14,23)(15,22)(16,21)(17,40)(18,39)(19,38)(20,37), (1,6,11,16)(2,15,12,5)(3,4,13,14)(7,20,17,10)(8,9,18,19)(21,33)(23,31)(24,40)(25,29)(26,38)(28,36)(30,34)(35,39), (2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,12)(21,28)(22,27)(23,26)(24,25)(29,40)(30,39)(31,38)(32,37)(33,36)(34,35) );`

`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)], [(1,36),(2,35),(3,34),(4,33),(5,32),(6,31),(7,30),(8,29),(9,28),(10,27),(11,26),(12,25),(13,24),(14,23),(15,22),(16,21),(17,40),(18,39),(19,38),(20,37)], [(1,6,11,16),(2,15,12,5),(3,4,13,14),(7,20,17,10),(8,9,18,19),(21,33),(23,31),(24,40),(25,29),(26,38),(28,36),(30,34),(35,39)], [(2,20),(3,19),(4,18),(5,17),(6,16),(7,15),(8,14),(9,13),(10,12),(21,28),(22,27),(23,26),(24,25),(29,40),(30,39),(31,38),(32,37),(33,36),(34,35)]])`

38 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E 4F 5A 5B 8A 8B 10A 10B 10C 10D 10E 10F 10G 10H 20A 20B 20C 20D 40A ··· 40H order 1 2 2 2 2 2 2 2 4 4 4 4 4 4 5 5 8 8 10 10 10 10 10 10 10 10 20 20 20 20 40 ··· 40 size 1 1 2 4 4 20 20 40 2 2 20 20 20 20 2 2 8 8 2 2 4 4 8 8 8 8 4 4 4 4 8 ··· 8

38 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 8 type + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 D4 D4 D4 D5 D10 D10 D20 D4⋊4D4 D4×D5 D20⋊1D4 kernel D20⋊1D4 D20⋊7C4 C5×C4.D4 C8⋊D10 C20⋊D4 D4⋊6D10 Dic10 D20 C22×C10 C4.D4 M4(2) C2×D4 C23 C5 C4 C1 # reps 1 2 1 2 1 1 2 2 2 2 4 2 8 2 4 2

Matrix representation of D201D4 in GL6(𝔽41)

 6 40 0 0 0 0 36 1 0 0 0 0 0 0 0 40 0 0 0 0 1 0 0 0 0 0 28 28 40 5 0 0 0 3 16 1
,
 2 11 0 0 0 0 37 39 0 0 0 0 0 0 0 0 1 0 0 0 28 28 40 5 0 0 1 0 0 0 0 0 32 32 0 13
,
 35 34 0 0 0 0 5 6 0 0 0 0 0 0 0 1 0 0 0 0 40 0 0 0 0 0 0 0 40 0 0 0 0 38 0 40
,
 35 34 0 0 0 0 5 6 0 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 0 0 28 28 40 5 0 0 3 0 0 1

`G:=sub<GL(6,GF(41))| [6,36,0,0,0,0,40,1,0,0,0,0,0,0,0,1,28,0,0,0,40,0,28,3,0,0,0,0,40,16,0,0,0,0,5,1],[2,37,0,0,0,0,11,39,0,0,0,0,0,0,0,28,1,32,0,0,0,28,0,32,0,0,1,40,0,0,0,0,0,5,0,13],[35,5,0,0,0,0,34,6,0,0,0,0,0,0,0,40,0,0,0,0,1,0,0,38,0,0,0,0,40,0,0,0,0,0,0,40],[35,5,0,0,0,0,34,6,0,0,0,0,0,0,40,0,28,3,0,0,0,1,28,0,0,0,0,0,40,0,0,0,0,0,5,1] >;`

D201D4 in GAP, Magma, Sage, TeX

`D_{20}\rtimes_1D_4`
`% in TeX`

`G:=Group("D20:1D4");`
`// GroupNames label`

`G:=SmallGroup(320,374);`
`// by ID`

`G=gap.SmallGroup(320,374);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,254,219,226,1123,570,136,1684,438,12550]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^20=b^2=c^4=d^2=1,b*a*b=d*a*d=a^-1,c*a*c^-1=a^9,c*b*c^-1=a^13*b,d*b*d=a^3*b,d*c*d=c^-1>;`
`// generators/relations`

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