Copied to
clipboard

G = D4○D8order 64 = 26

Central product of D4 and D8

p-group, metabelian, nilpotent (class 3), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C4 — D4○D8
 Chief series C1 — C2 — C4 — C2×C4 — C4○D4 — 2+ 1+4 — D4○D8
 Lower central C1 — C2 — C4 — D4○D8
 Upper central C1 — C2 — C4○D4 — D4○D8
 Jennings C1 — C2 — C2 — C4 — D4○D8

Generators and relations for D4○D8
G = < a,b,c,d | a4=b2=d2=1, c4=a2, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=a2c3 >

Subgroups: 237 in 134 conjugacy classes, 79 normal (9 characteristic)
C1, C2, C2 [×9], C4, C4 [×3], C4 [×2], C22 [×3], C22 [×12], C8, C8 [×3], C2×C4 [×3], C2×C4 [×6], D4 [×9], D4 [×12], Q8, Q8 [×2], C23 [×6], C2×C8 [×3], M4(2) [×3], D8 [×9], SD16 [×6], Q16, C2×D4 [×6], C2×D4 [×6], C4○D4, C4○D4 [×6], C4○D4 [×2], C8○D4, C2×D8 [×3], C4○D8 [×3], C8⋊C22 [×6], 2+ 1+4 [×2], D4○D8
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C24, C22×D4, D4○D8

Character table of D4○D8

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 4A 4B 4C 4D 4E 4F 8A 8B 8C 8D 8E size 1 1 2 2 2 4 4 4 4 4 4 2 2 2 2 4 4 2 2 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 trivial ρ2 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 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 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 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 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 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 linear of order 2 ρ9 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 ρ10 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 ρ11 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 ρ12 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 ρ13 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 ρ14 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 ρ15 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 ρ16 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 ρ17 2 2 -2 2 2 0 0 0 0 0 0 -2 -2 -2 2 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ18 2 2 2 2 -2 0 0 0 0 0 0 -2 -2 2 -2 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ19 2 2 2 -2 2 0 0 0 0 0 0 -2 2 -2 -2 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ20 2 2 -2 -2 -2 0 0 0 0 0 0 -2 2 2 2 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ21 4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2√2 -2√2 0 0 0 orthogonal faithful ρ22 4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -2√2 2√2 0 0 0 orthogonal faithful

Permutation representations of D4○D8
On 16 points - transitive group 16T80
Generators in S16
```(1 3 5 7)(2 4 6 8)(9 15 13 11)(10 16 14 12)
(1 14)(2 15)(3 16)(4 9)(5 10)(6 11)(7 12)(8 13)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(1 12)(2 11)(3 10)(4 9)(5 16)(6 15)(7 14)(8 13)```

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

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

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

`G:=TransitiveGroup(16,80);`

Matrix representation of D4○D8 in GL4(𝔽7) generated by

 0 6 2 6 5 0 2 1 1 1 5 5 5 2 1 2
,
 0 4 0 0 2 0 0 0 6 6 4 3 2 5 2 3
,
 2 0 5 1 1 2 2 1 1 6 6 5 5 5 1 5
,
 1 0 6 3 0 5 4 5 0 4 4 1 0 6 4 4
`G:=sub<GL(4,GF(7))| [0,5,1,5,6,0,1,2,2,2,5,1,6,1,5,2],[0,2,6,2,4,0,6,5,0,0,4,2,0,0,3,3],[2,1,1,5,0,2,6,5,5,2,6,1,1,1,5,5],[1,0,0,0,0,5,4,6,6,4,4,4,3,5,1,4] >;`

D4○D8 in GAP, Magma, Sage, TeX

`D_4\circ D_8`
`% in TeX`

`G:=Group("D4oD8");`
`// GroupNames label`

`G:=SmallGroup(64,257);`
`// by ID`

`G=gap.SmallGroup(64,257);`
`# by ID`

`G:=PCGroup([6,-2,2,2,2,-2,-2,217,255,1444,730,88]);`
`// Polycyclic`

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

Export

׿
×
𝔽