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G = D5⋊M4(2)  order 160 = 25·5

The semidirect product of D5 and M4(2) acting via M4(2)/C2×C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — D5⋊M4(2)
 Chief series C1 — C5 — C10 — Dic5 — C5⋊C8 — D5⋊C8 — D5⋊M4(2)
 Lower central C5 — C10 — D5⋊M4(2)
 Upper central C1 — C4 — C2×C4

Generators and relations for D5⋊M4(2)
G = < a,b,c,d | a5=b2=c8=d2=1, bab=a-1, cac-1=a3, ad=da, cbc-1=a2b, bd=db, dcd=c5 >

Subgroups: 196 in 68 conjugacy classes, 36 normal (24 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, C23, D5, D5, C10, C10, C2×C8, M4(2), C22×C4, Dic5, C20, D10, D10, C2×C10, C2×M4(2), C5⋊C8, C4×D5, C2×Dic5, C2×C20, C22×D5, D5⋊C8, C4.F5, C22.F5, C2×C4×D5, D5⋊M4(2)
Quotients: C1, C2, C4, C22, C2×C4, C23, M4(2), C22×C4, F5, C2×M4(2), C2×F5, C22×F5, D5⋊M4(2)

Character table of D5⋊M4(2)

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 5 8A 8B 8C 8D 8E 8F 8G 8H 10A 10B 10C 20A 20B 20C 20D size 1 1 2 5 5 10 1 1 2 5 5 10 4 10 10 10 10 10 10 10 10 4 4 4 4 4 4 4 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 -1 -1 -1 -1 -1 -1 1 1 1 1 -1 1 1 1 1 -1 -1 -1 1 1 1 -1 -1 -1 -1 linear of order 2 ρ3 1 1 -1 1 1 -1 1 1 -1 1 1 -1 1 1 -1 -1 1 1 -1 -1 1 -1 -1 1 1 1 -1 -1 linear of order 2 ρ4 1 1 -1 -1 -1 1 -1 -1 1 1 1 -1 1 -1 -1 -1 1 1 1 1 -1 -1 -1 1 -1 -1 1 1 linear of order 2 ρ5 1 1 1 -1 -1 -1 -1 -1 -1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 1 1 -1 -1 -1 -1 linear of order 2 ρ6 1 1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 1 1 1 1 1 1 1 linear of order 2 ρ7 1 1 -1 1 1 -1 1 1 -1 1 1 -1 1 -1 1 1 -1 -1 1 1 -1 -1 -1 1 1 1 -1 -1 linear of order 2 ρ8 1 1 -1 -1 -1 1 -1 -1 1 1 1 -1 1 1 1 1 -1 -1 -1 -1 1 -1 -1 1 -1 -1 1 1 linear of order 2 ρ9 1 1 -1 -1 -1 1 1 1 -1 -1 -1 1 1 i i -i -i i i -i -i -1 -1 1 1 1 -1 -1 linear of order 4 ρ10 1 1 -1 -1 -1 1 1 1 -1 -1 -1 1 1 -i -i i i -i -i i i -1 -1 1 1 1 -1 -1 linear of order 4 ρ11 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 1 i i -i i -i -i i -i 1 1 1 -1 -1 -1 -1 linear of order 4 ρ12 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 1 -i -i i -i i i -i i 1 1 1 -1 -1 -1 -1 linear of order 4 ρ13 1 1 -1 1 1 -1 -1 -1 1 -1 -1 1 1 -i i -i -i i -i i i -1 -1 1 -1 -1 1 1 linear of order 4 ρ14 1 1 -1 1 1 -1 -1 -1 1 -1 -1 1 1 i -i i i -i i -i -i -1 -1 1 -1 -1 1 1 linear of order 4 ρ15 1 1 1 -1 -1 -1 1 1 1 -1 -1 -1 1 -i i -i i -i i -i i 1 1 1 1 1 1 1 linear of order 4 ρ16 1 1 1 -1 -1 -1 1 1 1 -1 -1 -1 1 i -i i -i i -i i -i 1 1 1 1 1 1 1 linear of order 4 ρ17 2 -2 0 -2 2 0 2i -2i 0 -2i 2i 0 2 0 0 0 0 0 0 0 0 0 0 -2 2i -2i 0 0 complex lifted from M4(2) ρ18 2 -2 0 2 -2 0 -2i 2i 0 -2i 2i 0 2 0 0 0 0 0 0 0 0 0 0 -2 -2i 2i 0 0 complex lifted from M4(2) ρ19 2 -2 0 -2 2 0 -2i 2i 0 2i -2i 0 2 0 0 0 0 0 0 0 0 0 0 -2 -2i 2i 0 0 complex lifted from M4(2) ρ20 2 -2 0 2 -2 0 2i -2i 0 2i -2i 0 2 0 0 0 0 0 0 0 0 0 0 -2 2i -2i 0 0 complex lifted from M4(2) ρ21 4 4 -4 0 0 0 4 4 -4 0 0 0 -1 0 0 0 0 0 0 0 0 1 1 -1 -1 -1 1 1 orthogonal lifted from C2×F5 ρ22 4 4 4 0 0 0 -4 -4 -4 0 0 0 -1 0 0 0 0 0 0 0 0 -1 -1 -1 1 1 1 1 orthogonal lifted from C2×F5 ρ23 4 4 -4 0 0 0 -4 -4 4 0 0 0 -1 0 0 0 0 0 0 0 0 1 1 -1 1 1 -1 -1 orthogonal lifted from C2×F5 ρ24 4 4 4 0 0 0 4 4 4 0 0 0 -1 0 0 0 0 0 0 0 0 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from F5 ρ25 4 -4 0 0 0 0 -4i 4i 0 0 0 0 -1 0 0 0 0 0 0 0 0 -√5 √5 1 i -i -√-5 √-5 complex faithful ρ26 4 -4 0 0 0 0 -4i 4i 0 0 0 0 -1 0 0 0 0 0 0 0 0 √5 -√5 1 i -i √-5 -√-5 complex faithful ρ27 4 -4 0 0 0 0 4i -4i 0 0 0 0 -1 0 0 0 0 0 0 0 0 -√5 √5 1 -i i √-5 -√-5 complex faithful ρ28 4 -4 0 0 0 0 4i -4i 0 0 0 0 -1 0 0 0 0 0 0 0 0 √5 -√5 1 -i i -√-5 √-5 complex faithful

Smallest permutation representation of D5⋊M4(2)
On 40 points
Generators in S40
```(1 26 37 19 10)(2 20 27 11 38)(3 12 21 39 28)(4 40 13 29 22)(5 30 33 23 14)(6 24 31 15 34)(7 16 17 35 32)(8 36 9 25 18)
(1 10)(2 38)(3 28)(4 22)(5 14)(6 34)(7 32)(8 18)(11 20)(12 39)(15 24)(16 35)(19 26)(23 30)(25 36)(29 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)
(2 6)(4 8)(9 13)(11 15)(18 22)(20 24)(25 29)(27 31)(34 38)(36 40)```

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

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

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

Matrix representation of D5⋊M4(2) in GL4(𝔽41) generated by

 0 1 0 0 40 6 0 0 0 0 40 6 0 0 35 35
,
 0 1 0 0 1 0 0 0 0 0 40 6 0 0 0 1
,
 0 0 1 0 0 0 0 1 32 0 0 0 28 9 0 0
,
 1 0 0 0 0 1 0 0 0 0 40 0 0 0 0 40
`G:=sub<GL(4,GF(41))| [0,40,0,0,1,6,0,0,0,0,40,35,0,0,6,35],[0,1,0,0,1,0,0,0,0,0,40,0,0,0,6,1],[0,0,32,28,0,0,0,9,1,0,0,0,0,1,0,0],[1,0,0,0,0,1,0,0,0,0,40,0,0,0,0,40] >;`

D5⋊M4(2) in GAP, Magma, Sage, TeX

`D_5\rtimes M_4(2)`
`% in TeX`

`G:=Group("D5:M4(2)");`
`// GroupNames label`

`G:=SmallGroup(160,202);`
`// by ID`

`G=gap.SmallGroup(160,202);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-5,48,103,362,69,2309,599]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^5=b^2=c^8=d^2=1,b*a*b=a^-1,c*a*c^-1=a^3,a*d=d*a,c*b*c^-1=a^2*b,b*d=d*b,d*c*d=c^5>;`
`// generators/relations`

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