Copied to
clipboard

## G = C8.2D4order 64 = 26

### 2nd non-split extension by C8 of D4 acting via D4/C2=C22

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — C8.2D4
 Chief series C1 — C2 — C22 — C2×C4 — C42 — C8⋊C4 — C8.2D4
 Lower central C1 — C2 — C2×C4 — C8.2D4
 Upper central C1 — C22 — C42 — C8.2D4
 Jennings C1 — C2 — C2 — C2×C4 — C8.2D4

Generators and relations for C8.2D4
G = < a,b,c | a8=c2=1, b4=a4, bab-1=a5, cac=a3, cbc=b3 >

Subgroups: 113 in 62 conjugacy classes, 29 normal (11 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C42, C22⋊C4, C4⋊C4, C2×C8, SD16, Q16, C2×D4, C2×Q8, C2×Q8, C8⋊C4, C4.4D4, C4⋊Q8, C2×SD16, C2×Q16, C8.2D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C41D4, C8.C22, C8.2D4

Character table of C8.2D4

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 4F 4G 8A 8B 8C 8D size 1 1 1 1 8 2 2 4 4 8 8 8 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 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 linear of order 2 ρ7 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 linear of order 2 ρ9 2 -2 2 -2 0 2 -2 0 0 0 0 0 -2 0 2 0 orthogonal lifted from D4 ρ10 2 2 2 2 0 -2 -2 -2 2 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 -2 2 -2 0 -2 2 0 0 0 0 0 0 2 0 -2 orthogonal lifted from D4 ρ12 2 2 2 2 0 -2 -2 2 -2 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ13 2 -2 2 -2 0 2 -2 0 0 0 0 0 2 0 -2 0 orthogonal lifted from D4 ρ14 2 -2 2 -2 0 -2 2 0 0 0 0 0 0 -2 0 2 orthogonal lifted from D4 ρ15 4 -4 -4 4 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from C8.C22, Schur index 2 ρ16 4 4 -4 -4 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from C8.C22, Schur index 2

Smallest permutation representation of C8.2D4
On 32 points
Generators in S32
```(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)
(1 15 30 22 5 11 26 18)(2 12 31 19 6 16 27 23)(3 9 32 24 7 13 28 20)(4 14 25 21 8 10 29 17)
(1 30)(2 25)(3 28)(4 31)(5 26)(6 29)(7 32)(8 27)(10 12)(11 15)(14 16)(17 23)(19 21)(20 24)```

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

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

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

C8.2D4 is a maximal subgroup of
(C2×D4).D4  C42.5C23  C42.6C23  C42.7C23  M4(2)⋊10D4  C42.390C23  C42.409C23  C42.73C23  C42.75C23  C42.532C23
C42.D2p: C42.247D4  M4(2)⋊8D4  M4(2)⋊9D4  C42.256D4  C42.258D4  C42.273D4  C42.274D4  C42.276D4 ...
(C2×C8).D2p: M4(2).20D4  C24.31D4  C24.37D4  C40.31D4  C40.37D4  C56.31D4  C56.37D4 ...
D4.pD4⋊C2: C42.408C23  C42.411C23  SD166D4  D810D4  SD168D4  Q169D4  D8.13D4  C42.531C23 ...
C8.2D4 is a maximal quotient of
C42.D2p: C42.110D4  C42.111D4  C8.D12  C42.65D6  C42.80D6  C8.D20  C42.65D10  C42.80D10 ...
(C2×C8).D2p: C85SD16  C8.9SD16  C42.665C23  C42.666C23  C42.667C23  C8.2D8  C83Q16  C42.26Q8 ...

Matrix representation of C8.2D4 in GL6(𝔽17)

 0 1 0 0 0 0 16 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 4 0 0 0 13 0 0 0 0 4 0 0 0
,
 16 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 16 0 0 0
,
 1 0 0 0 0 0 0 16 0 0 0 0 0 0 0 16 0 0 0 0 16 0 0 0 0 0 0 0 1 0 0 0 0 0 0 16

`G:=sub<GL(6,GF(17))| [0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,13,0,0,0,4,0,0,0,0,0,0,4,0,0],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,16,0,0,0,0,0,0,0,16,0,0,0,0,16,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16] >;`

C8.2D4 in GAP, Magma, Sage, TeX

`C_8._2D_4`
`% in TeX`

`G:=Group("C8.2D4");`
`// GroupNames label`

`G:=SmallGroup(64,178);`
`// by ID`

`G=gap.SmallGroup(64,178);`
`# by ID`

`G:=PCGroup([6,-2,2,2,-2,2,-2,192,121,55,362,332,86,963,117]);`
`// Polycyclic`

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

Export

׿
×
𝔽