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G = C33⋊C2order 54 = 2·33

3rd semidirect product of C33 and C2 acting faithfully

metabelian, supersoluble, monomial, A-group, rational

Aliases: C333C2, C324S3, C3⋊(C3⋊S3), SmallGroup(54,14)

Series: Derived Chief Lower central Upper central

C1C33 — C33⋊C2
C1C3C32C33 — C33⋊C2
C33 — C33⋊C2
C1

Generators and relations for C33⋊C2
 G = < a,b,c,d | a3=b3=c3=d2=1, ab=ba, ac=ca, dad=a-1, bc=cb, dbd=b-1, dcd=c-1 >

Subgroups: 212 in 56 conjugacy classes, 29 normal (3 characteristic)
C1, C2, C3 [×13], S3 [×13], C32 [×13], C3⋊S3 [×13], C33, C33⋊C2
Quotients: C1, C2, S3 [×13], C3⋊S3 [×13], C33⋊C2

Character table of C33⋊C2

 class 123A3B3C3D3E3F3G3H3I3J3K3L3M
 size 1272222222222222
ρ1111111111111111    trivial
ρ21-11111111111111    linear of order 2
ρ320-1-122-1-1-1-12-12-1-1    orthogonal lifted from S3
ρ420-12-12-1-1-12-1-1-12-1    orthogonal lifted from S3
ρ520-1-1-1-1-1-1-1-1-12222    orthogonal lifted from S3
ρ620-12-1-1-122-1-1-12-1-1    orthogonal lifted from S3
ρ7202-1-12-1-12-1-12-1-1-1    orthogonal lifted from S3
ρ820-1-12-12-12-1-1-1-12-1    orthogonal lifted from S3
ρ9202-1-1-1-12-1-12-1-12-1    orthogonal lifted from S3
ρ1020-1-1-1-1-1-1222-1-1-12    orthogonal lifted from S3
ρ11202-1-1-12-1-12-1-12-1-1    orthogonal lifted from S3
ρ1220-1-1-1222-1-1-1-1-1-12    orthogonal lifted from S3
ρ1320222-1-1-1-1-1-1-1-1-12    orthogonal lifted from S3
ρ1420-1-12-1-12-12-12-1-1-1    orthogonal lifted from S3
ρ1520-12-1-12-1-1-122-1-1-1    orthogonal lifted from S3

Permutation representations of C33⋊C2
On 27 points - transitive group 27T7
Generators in S27
(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)
(1 6 14)(2 4 15)(3 5 13)(7 24 16)(8 22 17)(9 23 18)(10 27 19)(11 25 20)(12 26 21)
(1 26 8)(2 27 9)(3 25 7)(4 19 23)(5 20 24)(6 21 22)(10 18 15)(11 16 13)(12 17 14)
(2 3)(4 13)(5 15)(6 14)(7 27)(8 26)(9 25)(10 24)(11 23)(12 22)(16 19)(17 21)(18 20)

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

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

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

G:=TransitiveGroup(27,7);

Matrix representation of C33⋊C2 in GL6(ℤ)

010000
-1-10000
00-1-100
001000
0000-1-1
000010
,
100000
010000
00-1-100
001000
000001
0000-1-1
,
100000
010000
000100
00-1-100
000001
0000-1-1
,
100000
-1-10000
000-100
00-1000
0000-1-1
000001

G:=sub<GL(6,Integers())| [0,-1,0,0,0,0,1,-1,0,0,0,0,0,0,-1,1,0,0,0,0,-1,0,0,0,0,0,0,0,-1,1,0,0,0,0,-1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,1,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,1,-1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,0,0,0,0,1,-1,0,0,0,0,0,0,0,-1,0,0,0,0,1,-1],[1,-1,0,0,0,0,0,-1,0,0,0,0,0,0,0,-1,0,0,0,0,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,-1,1] >;

C33⋊C2 in GAP, Magma, Sage, TeX

C_3^3\rtimes C_2
% in TeX

G:=Group("C3^3:C2");
// GroupNames label

G:=SmallGroup(54,14);
// by ID

G=gap.SmallGroup(54,14);
# by ID

G:=PCGroup([4,-2,-3,-3,-3,33,146,579]);
// Polycyclic

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

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