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G = C34⋊4C4  order 324 = 22·34

4th semidirect product of C34 and C4 acting faithfully

metabelian, soluble, monomial, A-group

Aliases: C34⋊4C4, 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 C34⋊4C4
 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, C34⋊4C4
Quotients: C1, C2, C4, C32⋊C4, C34⋊4C4

Character table of C34⋊4C4

 class 123A3B3C3D3E3F3G3H3I3J3K3L3M3N3O3P3Q3R3S3T4A4B
 size 181444444444444444444448181
ρ1111111111111111111111111    trivial
ρ21111111111111111111111-1-1    linear of order 2
ρ31-111111111111111111111i-i    linear of order 4
ρ41-111111111111111111111-ii    linear of order 4
ρ5401-2-2-2-2-2-2-2-2-2411111111400    orthogonal lifted from C32⋊C4
ρ64044-2-211-21-21-2-2-211-21-21100    orthogonal lifted from C32⋊C4
ρ740-21-2-2114-21-21-211411-2-2-200    orthogonal lifted from C32⋊C4
ρ840-2114-21-21-2-21-21-21-2411-200    orthogonal lifted from C32⋊C4
ρ940-21-214-2-211-2-21111-2-24-2100    orthogonal lifted from C32⋊C4
ρ1040-21-2-2-2-2114111-21-2-2114-200    orthogonal lifted from C32⋊C4
ρ11401-24-2-21111-2-2-214-2-211-2100    orthogonal lifted from C32⋊C4
ρ12401-21111-2-24-2-2-21-211-2-24100    orthogonal lifted from C32⋊C4
ρ1340-21-21-211-2-24-2-241-21-211100    orthogonal lifted from C32⋊C4
ρ14401-2-241-21-211-21-21-214-2-2100    orthogonal lifted from C32⋊C4
ρ15401-21-21-2-2114114-21-21-2-2-200    orthogonal lifted from C32⋊C4
ρ1640-21111111114-2-2-2-2-2-2-2-2400    orthogonal lifted from C32⋊C4
ρ17401-211-2-241-21-21-2-24-2-211100    orthogonal lifted from C32⋊C4
ρ1840-21411-2-2-2-2111-2411-2-21-200    orthogonal lifted from C32⋊C4
ρ1940-211-21-214-2-2-211-2-241-21100    orthogonal lifted from C32⋊C4
ρ20404411-2-21-21-2111-2-21-21-2-200    orthogonal lifted from C32⋊C4
ρ21401-21-2411-2-211-2-2-2-21141-200    orthogonal lifted from C32⋊C4
ρ22401-2-21-21-24111-2-2114-21-2-200    orthogonal lifted from C32⋊C4
ρ2340-211-2-24-2-211-24-2-21111-2100    orthogonal lifted from C32⋊C4
ρ24401-2-211411-2-21411-2-2-2-21-200    orthogonal lifted from C32⋊C4

Permutation representations of C34⋊4C4
►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 C34⋊4C4 ►in GL8(ℤ)

-1-1000000
10000000
00100000
00010000
00001000
00000100
0000000-1
0000001-1
,
-1-1000000
10000000
00-1-10000
00100000
0000-1100
0000-1000
0000000-1
0000001-1
,
01000000
-1-1000000
00010000
00-1-10000
00001000
00000100
00000010
00000001
,
10000000
01000000
00-1-10000
00100000
00001000
00000100
00000010
00000001
,
00100000
00010000
01000000
10000000
00000010
00000001
00000100
00001000

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] >;
 

C34⋊4C4 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

Character table of C34⋊4C4 in TeX

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