Copied to
clipboard

## G = C8.12D4order 64 = 26

### 8th non-split extension by C8 of D4 acting via D4/C4=C2

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — C8.12D4
 Chief series C1 — C2 — C22 — C2×C4 — C42 — C4×C8 — C8.12D4
 Lower central C1 — C2 — C2×C4 — C8.12D4
 Upper central C1 — C22 — C42 — C8.12D4
 Jennings C1 — C2 — C2 — C2×C4 — C8.12D4

Generators and relations for C8.12D4
G = < a,b,c | a8=b4=1, c2=a4, ab=ba, cac-1=a3, cbc-1=a4b-1 >

Subgroups: 129 in 65 conjugacy classes, 29 normal (13 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C42, C22⋊C4, C2×C8, D8, SD16, Q16, C2×D4, C2×Q8, C4×C8, C4.4D4, C2×D8, C2×SD16, C2×Q16, C8.12D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C41D4, C4○D8, C8.12D4

Character table of C8.12D4

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G 4H 8A 8B 8C 8D 8E 8F 8G 8H size 1 1 1 1 8 8 2 2 2 2 2 2 8 8 2 2 2 2 2 2 2 2 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 2 -2 -2 2 0 0 0 0 2 0 -2 0 0 0 -2 0 2 0 2 0 -2 0 orthogonal lifted from D4 ρ10 2 2 2 2 0 0 2 -2 -2 -2 -2 2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 -2 -2 2 0 0 0 0 2 0 -2 0 0 0 2 0 -2 0 -2 0 2 0 orthogonal lifted from D4 ρ12 2 -2 -2 2 0 0 0 0 -2 0 2 0 0 0 0 2 0 -2 0 -2 0 2 orthogonal lifted from D4 ρ13 2 -2 -2 2 0 0 0 0 -2 0 2 0 0 0 0 -2 0 2 0 2 0 -2 orthogonal lifted from D4 ρ14 2 2 2 2 0 0 -2 2 -2 2 -2 -2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ15 2 -2 2 -2 0 0 0 -2i 0 2i 0 0 0 0 -√-2 -√2 √-2 √2 -√-2 -√2 √-2 √2 complex lifted from C4○D8 ρ16 2 2 -2 -2 0 0 -2i 0 0 0 0 2i 0 0 √-2 -√2 √-2 -√2 -√-2 √2 -√-2 √2 complex lifted from C4○D8 ρ17 2 -2 2 -2 0 0 0 2i 0 -2i 0 0 0 0 √-2 -√2 -√-2 √2 √-2 -√2 -√-2 √2 complex lifted from C4○D8 ρ18 2 2 -2 -2 0 0 2i 0 0 0 0 -2i 0 0 √-2 √2 √-2 √2 -√-2 -√2 -√-2 -√2 complex lifted from C4○D8 ρ19 2 2 -2 -2 0 0 2i 0 0 0 0 -2i 0 0 -√-2 -√2 -√-2 -√2 √-2 √2 √-2 √2 complex lifted from C4○D8 ρ20 2 2 -2 -2 0 0 -2i 0 0 0 0 2i 0 0 -√-2 √2 -√-2 √2 √-2 -√2 √-2 -√2 complex lifted from C4○D8 ρ21 2 -2 2 -2 0 0 0 2i 0 -2i 0 0 0 0 -√-2 √2 √-2 -√2 -√-2 √2 √-2 -√2 complex lifted from C4○D8 ρ22 2 -2 2 -2 0 0 0 -2i 0 2i 0 0 0 0 √-2 √2 -√-2 -√2 √-2 √2 -√-2 -√2 complex lifted from C4○D8

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

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

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

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

C8.12D4 is a maximal subgroup of
C42.410C23  C42.411C23  C42.533C23
C8.D4p: C8.30D8  C8.3D8  C8.21D8  C163D4  C8.7D8  C8.8D12  C8.8D20  C8.8D28 ...
C42.D2p: D8.5D4  Q16.5D4  C8.22SD16  C8.12SD16  C42.360D4  M4(2)⋊9D4  C42.308D4  C42.260D4 ...
(C2p×Q16)⋊C2: C42.387C23  Q164D4  Q1612D4  C42.527C23  C42.530C23  C42.532C23  C24.28D4  C40.28D4 ...
(C2p×SD16)⋊C2: C42.385C23  SD162D4  SD1610D4  C42.528C23  C42.531C23  C24.43D4  C40.43D4  C56.43D4 ...
C4p.(C2×D4): M4(2)⋊10D4  M4(2)⋊11D4  M4(2).20D4  D85D4  D812D4  D8○SD16  C24.22D4  C40.22D4 ...
C8.12D4 is a maximal quotient of
C42.D2p: C42.433D4  C8.8D12  C42.214D6  C8.8D20  C42.214D10  C8.8D28  C42.214D14 ...
(C2×D4).D2p: (C22×D8).C2  (C2×C8).41D4  C24.22D4  C24.43D4  C40.22D4  C40.43D4  C56.22D4  C56.43D4 ...
(C2×C8).D2p: C85D8  C85Q16  C8212C2  C825C2  C8.7Q16  C823C2  C42.664C23  C42.665C23 ...

Matrix representation of C8.12D4 in GL4(𝔽17) generated by

 12 5 0 0 12 12 0 0 0 0 12 5 0 0 12 12
,
 0 13 0 0 4 0 0 0 0 0 4 0 0 0 0 4
,
 0 13 0 0 13 0 0 0 0 0 4 0 0 0 0 13
`G:=sub<GL(4,GF(17))| [12,12,0,0,5,12,0,0,0,0,12,12,0,0,5,12],[0,4,0,0,13,0,0,0,0,0,4,0,0,0,0,4],[0,13,0,0,13,0,0,0,0,0,4,0,0,0,0,13] >;`

C8.12D4 in GAP, Magma, Sage, TeX

`C_8._{12}D_4`
`% in TeX`

`G:=Group("C8.12D4");`
`// GroupNames label`

`G:=SmallGroup(64,176);`
`// by ID`

`G=gap.SmallGroup(64,176);`
`# by ID`

`G:=PCGroup([6,-2,2,2,-2,2,-2,121,55,362,230,963,117]);`
`// Polycyclic`

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

Export

׿
×
𝔽