Copied to
clipboard

## G = C42.12C4order 64 = 26

### 9th non-split extension by C42 of C4 acting via C4/C2=C2

p-group, metabelian, nilpotent (class 2), monomial

Series: Derived Chief Lower central Upper central Jennings

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

Generators and relations for C42.12C4
G = < a,b,c | a4=b4=1, c4=a2, ab=ba, cac-1=a-1b2, bc=cb >

Subgroups: 73 in 59 conjugacy classes, 45 normal (21 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, C23, C42, C2×C8, C22×C4, C4×C8, C22⋊C8, C4⋊C8, C2×C42, C42.12C4
Quotients: C1, C2, C4, C22, C8, C2×C4, C23, C2×C8, M4(2), C22×C4, C4○D4, C42⋊C2, C22×C8, C2×M4(2), C42.12C4

Smallest permutation representation of C42.12C4
On 32 points
Generators in S32
```(1 3 5 7)(2 26 6 30)(4 28 8 32)(9 11 13 15)(10 24 14 20)(12 18 16 22)(17 19 21 23)(25 27 29 31)
(1 21 27 13)(2 22 28 14)(3 23 29 15)(4 24 30 16)(5 17 31 9)(6 18 32 10)(7 19 25 11)(8 20 26 12)
(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)```

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

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

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

40 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A ··· 4L 4M ··· 4R 8A ··· 8P order 1 2 2 2 2 2 4 ··· 4 4 ··· 4 8 ··· 8 size 1 1 1 1 2 2 1 ··· 1 2 ··· 2 2 ··· 2

40 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 type + + + + + image C1 C2 C2 C2 C2 C4 C4 C8 M4(2) C4○D4 kernel C42.12C4 C4×C8 C22⋊C8 C4⋊C8 C2×C42 C42 C22×C4 C2×C4 C4 C4 # reps 1 2 2 2 1 4 4 16 4 4

Matrix representation of C42.12C4 in GL3(𝔽17) generated by

 13 0 0 0 16 0 0 4 1
,
 13 0 0 0 13 0 0 0 13
,
 15 0 0 0 13 15 0 0 4
`G:=sub<GL(3,GF(17))| [13,0,0,0,16,4,0,0,1],[13,0,0,0,13,0,0,0,13],[15,0,0,0,13,0,0,15,4] >;`

C42.12C4 in GAP, Magma, Sage, TeX

`C_4^2._{12}C_4`
`% in TeX`

`G:=Group("C4^2.12C4");`
`// GroupNames label`

`G:=SmallGroup(64,112);`
`// by ID`

`G=gap.SmallGroup(64,112);`
`# by ID`

`G:=PCGroup([6,-2,2,2,-2,2,-2,96,121,50,88]);`
`// Polycyclic`

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

׿
×
𝔽