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G = C344C4order 324 = 22·34

4th semidirect product of C34 and C4 acting faithfully

metabelian, soluble, monomial, A-group

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

Series: Derived Chief Lower central Upper central

C1C34 — C344C4
C1C32C34C34⋊C2 — C344C4
C34 — C344C4
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 [×20], C4, S3 [×20], C32 [×10], C32 [×60], C3⋊S3 [×70], C33 [×20], C32⋊C4 [×10], C33⋊C2 [×20], C34, C34⋊C2, C344C4
Quotients: C1, C2, C4, C32⋊C4 [×10], C344C4

Character table of C344C4

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

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

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

-1-1000000
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] >;

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

Character table of C344C4 in TeX

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