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G = D8⋊2C4order 64 = 26

2nd semidirect product of D8 and C4 acting via C4/C2=C2

p-group, metabelian, nilpotent (class 4), monomial

Aliases: D82C4, Q162C4, C8.22D4, C4.8SD16, C22.3D8, M5(2)⋊5C2, C8.1(C2×C4), C4.Q81C2, C4○D8.2C2, (C2×C4).10D4, (C2×C8).9C22, C4.4(C22⋊C4), C2.9(D4⋊C4), SmallGroup(64,41)

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C8 — D8⋊2C4
 Chief series C1 — C2 — C4 — C2×C4 — C2×C8 — C4○D8 — D8⋊2C4
 Lower central C1 — C2 — C4 — C8 — D8⋊2C4
 Upper central C1 — C2 — C2×C4 — C2×C8 — D8⋊2C4
 Jennings C1 — C2 — C2 — C2 — C2 — C4 — C4 — C2×C8 — D8⋊2C4

Generators and relations for D82C4
G = < a,b,c | a8=b2=c4=1, bab=a-1, cac-1=a3, cbc-1=a5b >

Character table of D82C4

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 8A 8B 8C 16A 16B 16C 16D size 1 1 2 8 2 2 8 8 8 2 2 4 4 4 4 4 ρ1 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 linear of order 2 ρ3 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 linear of order 2 ρ5 1 1 -1 1 1 -1 -i i -1 -1 -1 1 i -i i -i linear of order 4 ρ6 1 1 -1 1 1 -1 i -i -1 -1 -1 1 -i i -i i linear of order 4 ρ7 1 1 -1 -1 1 -1 i -i 1 -1 -1 1 i -i i -i linear of order 4 ρ8 1 1 -1 -1 1 -1 -i i 1 -1 -1 1 -i i -i i linear of order 4 ρ9 2 2 -2 0 2 -2 0 0 0 2 2 -2 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 2 0 2 2 0 0 0 -2 -2 -2 0 0 0 0 orthogonal lifted from D4 ρ11 2 2 2 0 -2 -2 0 0 0 0 0 0 -√2 √2 √2 -√2 orthogonal lifted from D8 ρ12 2 2 2 0 -2 -2 0 0 0 0 0 0 √2 -√2 -√2 √2 orthogonal lifted from D8 ρ13 2 2 -2 0 -2 2 0 0 0 0 0 0 √-2 √-2 -√-2 -√-2 complex lifted from SD16 ρ14 2 2 -2 0 -2 2 0 0 0 0 0 0 -√-2 -√-2 √-2 √-2 complex lifted from SD16 ρ15 4 -4 0 0 0 0 0 0 0 -2√-2 2√-2 0 0 0 0 0 complex faithful ρ16 4 -4 0 0 0 0 0 0 0 2√-2 -2√-2 0 0 0 0 0 complex faithful

Permutation representations of D82C4
On 16 points - transitive group 16T156
Generators in S16
```(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(1 16)(2 15)(3 14)(4 13)(5 12)(6 11)(7 10)(8 9)
(2 4)(3 7)(6 8)(9 10 13 14)(11 16 15 12)```

`G:=sub<Sym(16)| (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,16)(2,15)(3,14)(4,13)(5,12)(6,11)(7,10)(8,9), (2,4)(3,7)(6,8)(9,10,13,14)(11,16,15,12)>;`

`G:=Group( (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,16)(2,15)(3,14)(4,13)(5,12)(6,11)(7,10)(8,9), (2,4)(3,7)(6,8)(9,10,13,14)(11,16,15,12) );`

`G=PermutationGroup([(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16)], [(1,16),(2,15),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9)], [(2,4),(3,7),(6,8),(9,10,13,14),(11,16,15,12)])`

`G:=TransitiveGroup(16,156);`

D82C4 is a maximal subgroup of
C23.13D8  Q32⋊C4  D8⋊D4  D8.D4  D10.D8
D8p⋊C4: D16⋊C4  D248C4  D242C4  D4014C4  D408C4  D401C4  D568C4  D562C4 ...
C4p.SD16: D83Q8  D8.2Q8  D82Dic3  D82Dic5  D82Dic7 ...
D82C4 is a maximal quotient of
D8⋊C8  Q16⋊C8  C22.SD32  C23.32D8  C8.C42  D10.D8  D401C4
C8.D4p: C8.30D8  D248C4  D4014C4  D568C4 ...
C4p.SD16: C8.16Q16  D242C4  D82Dic3  D408C4  D82Dic5  D562C4  D82Dic7 ...

Matrix representation of D82C4 in GL4(𝔽3) generated by

 2 0 0 2 0 0 1 0 0 1 1 0 2 0 0 0
,
 0 0 2 0 1 0 0 1 2 0 0 0 0 1 1 0
,
 1 0 0 2 0 0 2 0 0 1 0 0 0 0 0 2
`G:=sub<GL(4,GF(3))| [2,0,0,2,0,0,1,0,0,1,1,0,2,0,0,0],[0,1,2,0,0,0,0,1,2,0,0,1,0,1,0,0],[1,0,0,0,0,0,1,0,0,2,0,0,2,0,0,2] >;`

D82C4 in GAP, Magma, Sage, TeX

`D_8\rtimes_2C_4`
`% in TeX`

`G:=Group("D8:2C4");`
`// GroupNames label`

`G:=SmallGroup(64,41);`
`// by ID`

`G=gap.SmallGroup(64,41);`
`# by ID`

`G:=PCGroup([6,-2,2,-2,2,-2,-2,48,73,362,476,86,489,1444,730,88]);`
`// Polycyclic`

`G:=Group<a,b,c|a^8=b^2=c^4=1,b*a*b=a^-1,c*a*c^-1=a^3,c*b*c^-1=a^5*b>;`
`// generators/relations`

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