Copied to
clipboard

## G = C34⋊4C4order 324 = 22·34

### 4th semidirect product of C34 and C4 acting faithfully

Aliases: C344C4, C32⋊(C32⋊C4), C34⋊C2.C2, SmallGroup(324,164)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C34 — C34⋊4C4
 Chief series C1 — C32 — C34 — C34⋊C2 — C34⋊4C4
 Lower central C34 — C34⋊4C4
 Upper central C1

Generators and relations for C344C4
G = < a,b,c,d,e | a3=b3=c3=d3=e4=1, ebe-1=ab=ba, ac=ca, ad=da, eae-1=a-1b, bc=cb, bd=db, ede-1=cd=dc, ece-1=c-1d >

Subgroups: 2836 in 236 conjugacy classes, 14 normal (4 characteristic)
C1, C2, C3, C4, S3, C32, C32, C3⋊S3, C33, C32⋊C4, C33⋊C2, C34, C34⋊C2, C344C4
Quotients: C1, C2, C4, C32⋊C4, C344C4

Character table of C344C4

 class 1 2 3A 3B 3C 3D 3E 3F 3G 3H 3I 3J 3K 3L 3M 3N 3O 3P 3Q 3R 3S 3T 4A 4B size 1 81 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 81 81 ρ1 1 1 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 -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 i -i linear of order 4 ρ4 1 -1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 -i i linear of order 4 ρ5 4 0 1 -2 -2 -2 -2 -2 -2 -2 -2 -2 4 1 1 1 1 1 1 1 1 4 0 0 orthogonal lifted from C32⋊C4 ρ6 4 0 4 4 -2 -2 1 1 -2 1 -2 1 -2 -2 -2 1 1 -2 1 -2 1 1 0 0 orthogonal lifted from C32⋊C4 ρ7 4 0 -2 1 -2 -2 1 1 4 -2 1 -2 1 -2 1 1 4 1 1 -2 -2 -2 0 0 orthogonal lifted from C32⋊C4 ρ8 4 0 -2 1 1 4 -2 1 -2 1 -2 -2 1 -2 1 -2 1 -2 4 1 1 -2 0 0 orthogonal lifted from C32⋊C4 ρ9 4 0 -2 1 -2 1 4 -2 -2 1 1 -2 -2 1 1 1 1 -2 -2 4 -2 1 0 0 orthogonal lifted from C32⋊C4 ρ10 4 0 -2 1 -2 -2 -2 -2 1 1 4 1 1 1 -2 1 -2 -2 1 1 4 -2 0 0 orthogonal lifted from C32⋊C4 ρ11 4 0 1 -2 4 -2 -2 1 1 1 1 -2 -2 -2 1 4 -2 -2 1 1 -2 1 0 0 orthogonal lifted from C32⋊C4 ρ12 4 0 1 -2 1 1 1 1 -2 -2 4 -2 -2 -2 1 -2 1 1 -2 -2 4 1 0 0 orthogonal lifted from C32⋊C4 ρ13 4 0 -2 1 -2 1 -2 1 1 -2 -2 4 -2 -2 4 1 -2 1 -2 1 1 1 0 0 orthogonal lifted from C32⋊C4 ρ14 4 0 1 -2 -2 4 1 -2 1 -2 1 1 -2 1 -2 1 -2 1 4 -2 -2 1 0 0 orthogonal lifted from C32⋊C4 ρ15 4 0 1 -2 1 -2 1 -2 -2 1 1 4 1 1 4 -2 1 -2 1 -2 -2 -2 0 0 orthogonal lifted from C32⋊C4 ρ16 4 0 -2 1 1 1 1 1 1 1 1 1 4 -2 -2 -2 -2 -2 -2 -2 -2 4 0 0 orthogonal lifted from C32⋊C4 ρ17 4 0 1 -2 1 1 -2 -2 4 1 -2 1 -2 1 -2 -2 4 -2 -2 1 1 1 0 0 orthogonal lifted from C32⋊C4 ρ18 4 0 -2 1 4 1 1 -2 -2 -2 -2 1 1 1 -2 4 1 1 -2 -2 1 -2 0 0 orthogonal lifted from C32⋊C4 ρ19 4 0 -2 1 1 -2 1 -2 1 4 -2 -2 -2 1 1 -2 -2 4 1 -2 1 1 0 0 orthogonal lifted from C32⋊C4 ρ20 4 0 4 4 1 1 -2 -2 1 -2 1 -2 1 1 1 -2 -2 1 -2 1 -2 -2 0 0 orthogonal lifted from C32⋊C4 ρ21 4 0 1 -2 1 -2 4 1 1 -2 -2 1 1 -2 -2 -2 -2 1 1 4 1 -2 0 0 orthogonal lifted from C32⋊C4 ρ22 4 0 1 -2 -2 1 -2 1 -2 4 1 1 1 -2 -2 1 1 4 -2 1 -2 -2 0 0 orthogonal lifted from C32⋊C4 ρ23 4 0 -2 1 1 -2 -2 4 -2 -2 1 1 -2 4 -2 -2 1 1 1 1 -2 1 0 0 orthogonal lifted from C32⋊C4 ρ24 4 0 1 -2 -2 1 1 4 1 1 -2 -2 1 4 1 1 -2 -2 -2 -2 1 -2 0 0 orthogonal lifted from C32⋊C4

Permutation representations of C344C4
On 18 points - transitive group 18T128
Generators in S18
```(1 7 9)(2 10 8)(3 14 17)(4 18 11)(5 15 12)(6 13 16)
(2 8 10)(4 11 18)(6 16 13)
(1 14 12)(3 15 9)(5 7 17)
(1 14 12)(2 13 11)(3 15 9)(4 10 16)(5 7 17)(6 18 8)
(1 2)(3 4 5 6)(7 8 9 10)(11 12 13 14)(15 16 17 18)```

`G:=sub<Sym(18)| (1,7,9)(2,10,8)(3,14,17)(4,18,11)(5,15,12)(6,13,16), (2,8,10)(4,11,18)(6,16,13), (1,14,12)(3,15,9)(5,7,17), (1,14,12)(2,13,11)(3,15,9)(4,10,16)(5,7,17)(6,18,8), (1,2)(3,4,5,6)(7,8,9,10)(11,12,13,14)(15,16,17,18)>;`

`G:=Group( (1,7,9)(2,10,8)(3,14,17)(4,18,11)(5,15,12)(6,13,16), (2,8,10)(4,11,18)(6,16,13), (1,14,12)(3,15,9)(5,7,17), (1,14,12)(2,13,11)(3,15,9)(4,10,16)(5,7,17)(6,18,8), (1,2)(3,4,5,6)(7,8,9,10)(11,12,13,14)(15,16,17,18) );`

`G=PermutationGroup([[(1,7,9),(2,10,8),(3,14,17),(4,18,11),(5,15,12),(6,13,16)], [(2,8,10),(4,11,18),(6,16,13)], [(1,14,12),(3,15,9),(5,7,17)], [(1,14,12),(2,13,11),(3,15,9),(4,10,16),(5,7,17),(6,18,8)], [(1,2),(3,4,5,6),(7,8,9,10),(11,12,13,14),(15,16,17,18)]])`

`G:=TransitiveGroup(18,128);`

Matrix representation of C344C4 in GL8(ℤ)

 -1 -1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 1 -1
,
 -1 -1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 -1 -1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 -1 1 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 1 -1
,
 0 1 0 0 0 0 0 0 -1 -1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 -1 -1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 -1 -1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0

`G:=sub<GL(8,Integers())| [-1,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,-1],[-1,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,-1],[0,-1,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0] >;`

C344C4 in GAP, Magma, Sage, TeX

`C_3^4\rtimes_4C_4`
`% in TeX`

`G:=Group("C3^4:4C4");`
`// GroupNames label`

`G:=SmallGroup(324,164);`
`// by ID`

`G=gap.SmallGroup(324,164);`
`# by ID`

`G:=PCGroup([6,-2,-2,-3,3,-3,3,12,506,80,771,297,7564,1090,10373,3899]);`
`// Polycyclic`

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

Export

׿
×
𝔽