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G = C22⋊D20order 160 = 25·5

The semidirect product of C22 and D20 acting via D20/D10=C2

Series: Derived Chief Lower central Upper central

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

Generators and relations for C22⋊D20
G = < a,b,c,d | a2=b2=c20=d2=1, cac-1=dad=ab=ba, bc=cb, bd=db, dcd=c-1 >

Subgroups: 544 in 130 conjugacy classes, 37 normal (15 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C5, C2×C4, C2×C4, D4, C23, C23, D5, C10, C10, C10, C22⋊C4, C22⋊C4, C2×D4, C24, Dic5, C20, D10, D10, C2×C10, C2×C10, C2×C10, C22≀C2, D20, C2×Dic5, C5⋊D4, C2×C20, C22×D5, C22×D5, C22×D5, C22×C10, D10⋊C4, C5×C22⋊C4, C2×D20, C2×C5⋊D4, C23×D5, C22⋊D20
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, D10, C22≀C2, D20, C22×D5, C2×D20, D4×D5, C22⋊D20

Smallest permutation representation of C22⋊D20
On 40 points
Generators in S40
```(2 36)(4 38)(6 40)(8 22)(10 24)(12 26)(14 28)(16 30)(18 32)(20 34)
(1 35)(2 36)(3 37)(4 38)(5 39)(6 40)(7 21)(8 22)(9 23)(10 24)(11 25)(12 26)(13 27)(14 28)(15 29)(16 30)(17 31)(18 32)(19 33)(20 34)
(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 34)(2 33)(3 32)(4 31)(5 30)(6 29)(7 28)(8 27)(9 26)(10 25)(11 24)(12 23)(13 22)(14 21)(15 40)(16 39)(17 38)(18 37)(19 36)(20 35)```

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

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

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

34 conjugacy classes

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

34 irreducible representations

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

Matrix representation of C22⋊D20 in GL4(𝔽41) generated by

 1 0 0 0 0 1 0 0 0 0 1 0 0 0 21 40
,
 1 0 0 0 0 1 0 0 0 0 40 0 0 0 0 40
,
 9 11 0 0 30 14 0 0 0 0 1 37 0 0 0 40
,
 32 30 0 0 11 9 0 0 0 0 40 4 0 0 0 1
`G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,1,21,0,0,0,40],[1,0,0,0,0,1,0,0,0,0,40,0,0,0,0,40],[9,30,0,0,11,14,0,0,0,0,1,0,0,0,37,40],[32,11,0,0,30,9,0,0,0,0,40,0,0,0,4,1] >;`

C22⋊D20 in GAP, Magma, Sage, TeX

`C_2^2\rtimes D_{20}`
`% in TeX`

`G:=Group("C2^2:D20");`
`// GroupNames label`

`G:=SmallGroup(160,103);`
`// by ID`

`G=gap.SmallGroup(160,103);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-5,218,188,50,4613]);`
`// Polycyclic`

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

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