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G = C33⋊C4order 108 = 22·33

2nd semidirect product of C33 and C4 acting faithfully

metabelian, soluble, monomial, A-group

Aliases: C332C4, C323Dic3, C3⋊S3.S3, C3⋊(C32⋊C4), (C3×C3⋊S3).2C2, SmallGroup(108,37)

Series: Derived Chief Lower central Upper central

C1C33 — C33⋊C4
C1C3C33C3×C3⋊S3 — C33⋊C4
C33 — C33⋊C4
C1

Generators and relations for C33⋊C4
 G = < a,b,c,d | a3=b3=c3=d4=1, ab=ba, ac=ca, dad-1=ab-1, bc=cb, dbd-1=a-1b-1, dcd-1=c-1 >

9C2
2C3
2C3
4C3
4C3
27C4
6S3
6S3
9C6
2C32
2C32
4C32
4C32
9Dic3
6C3×S3
6C3×S3
3C32⋊C4

Character table of C33⋊C4

 class 123A3B3C3D3E3F3G4A4B6
 size 192444444272718
ρ1111111111111    trivial
ρ2111111111-1-11    linear of order 2
ρ31-11111111i-i-1    linear of order 4
ρ41-11111111-ii-1    linear of order 4
ρ522-1-1-122-1-100-1    orthogonal lifted from S3
ρ62-2-1-1-122-1-1001    symplectic lifted from Dic3, Schur index 2
ρ74041-21-21-2000    orthogonal lifted from C32⋊C4
ρ8404-21-21-21000    orthogonal lifted from C32⋊C4
ρ940-21-1+3-3/2-211-1-3-3/2000    complex faithful
ρ1040-2-1+3-3/211-2-1-3-3/21000    complex faithful
ρ1140-21-1-3-3/2-211-1+3-3/2000    complex faithful
ρ1240-2-1-3-3/211-2-1+3-3/21000    complex faithful

Permutation representations of C33⋊C4
On 12 points - transitive group 12T72
Generators in S12
(1 8 9)(2 5 10)(3 11 6)(4 12 7)
(2 10 5)(4 7 12)
(1 9 8)(2 5 10)(3 11 6)(4 7 12)
(1 2 3 4)(5 6 7 8)(9 10 11 12)

G:=sub<Sym(12)| (1,8,9)(2,5,10)(3,11,6)(4,12,7), (2,10,5)(4,7,12), (1,9,8)(2,5,10)(3,11,6)(4,7,12), (1,2,3,4)(5,6,7,8)(9,10,11,12)>;

G:=Group( (1,8,9)(2,5,10)(3,11,6)(4,12,7), (2,10,5)(4,7,12), (1,9,8)(2,5,10)(3,11,6)(4,7,12), (1,2,3,4)(5,6,7,8)(9,10,11,12) );

G=PermutationGroup([(1,8,9),(2,5,10),(3,11,6),(4,12,7)], [(2,10,5),(4,7,12)], [(1,9,8),(2,5,10),(3,11,6),(4,7,12)], [(1,2,3,4),(5,6,7,8),(9,10,11,12)])

G:=TransitiveGroup(12,72);

On 18 points - transitive group 18T54
Generators in S18
(1 17 15)(4 7 9)(5 14 12)
(1 17 15)(2 18 16)(3 8 10)(4 7 9)(5 14 12)(6 11 13)
(1 4 5)(2 6 3)(7 14 17)(8 18 11)(9 12 15)(10 16 13)
(1 2)(3 4)(5 6)(7 8 9 10)(11 12 13 14)(15 16 17 18)

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

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

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

G:=TransitiveGroup(18,54);

On 27 points - transitive group 27T31
Generators in S27
(1 8 10)(2 22 20)(3 14 12)(4 23 16)(5 13 19)(6 18 21)(7 17 15)(9 24 25)(11 27 26)
(1 24 26)(2 7 5)(3 16 18)(4 21 14)(6 12 23)(8 25 11)(9 27 10)(13 22 17)(15 19 20)
(1 3 2)(4 17 25)(5 26 18)(6 19 27)(7 24 16)(8 14 22)(9 23 15)(10 12 20)(11 21 13)
(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)

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

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

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

G:=TransitiveGroup(27,31);

C33⋊C4 is a maximal subgroup of   S3×C32⋊C4  C33⋊D4  C33⋊Q8  C323Dic9  C34⋊C4  C62⋊Dic3
C33⋊C4 is a maximal quotient of   C334C8  C323Dic9  He34Dic3  C34⋊C4  C62⋊Dic3

Polynomial with Galois group C33⋊C4 over ℚ
actionf(x)Disc(f)
12T72x12+12x10-40x9+414x8-1416x7+3388x6-6552x5+8001x4-7448x3+7056x2+4704x+2744287·316·714·172·13612·15427272

Matrix representation of C33⋊C4 in GL4(𝔽7) generated by

6211
2661
0010
0002
,
6251
0405
4406
0002
,
3145
1335
0040
0002
,
1314
3645
1632
5514
G:=sub<GL(4,GF(7))| [6,2,0,0,2,6,0,0,1,6,1,0,1,1,0,2],[6,0,4,0,2,4,4,0,5,0,0,0,1,5,6,2],[3,1,0,0,1,3,0,0,4,3,4,0,5,5,0,2],[1,3,1,5,3,6,6,5,1,4,3,1,4,5,2,4] >;

C33⋊C4 in GAP, Magma, Sage, TeX

C_3^3\rtimes C_4
% in TeX

G:=Group("C3^3:C4");
// GroupNames label

G:=SmallGroup(108,37);
// by ID

G=gap.SmallGroup(108,37);
# by ID

G:=PCGroup([5,-2,-2,-3,3,-3,10,302,67,323,248,1804]);
// Polycyclic

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

Export

Subgroup lattice of C33⋊C4 in TeX
Character table of C33⋊C4 in TeX

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