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G = C33⋊A4order 324 = 22·34

The semidirect product of C33 and A4 acting faithfully

non-abelian, soluble, monomial

Aliases: C33⋊A4, C324D6⋊C3, SmallGroup(324,160)

Series: Derived Chief Lower central Upper central

C1C33C324D6 — C33⋊A4
C1C33C324D6 — C33⋊A4
C324D6 — C33⋊A4
C1

Generators and relations for C33⋊A4
 G = < a,b,c,d,e,f | a3=b3=c3=d2=e2=f3=1, ab=ba, ac=ca, dad=a-1, ae=ea, faf-1=c, bc=cb, dbd=ebe=b-1, fbf-1=a, cd=dc, ece=c-1, fcf-1=b, fdf-1=de=ed, fef-1=d >

27C2
3C3
4C3
6C3
36C3
27C22
9S3
9S3
18S3
27C6
3C32
4C32
6C32
12C9
12C32
12C9
27D6
27A4
3C3⋊S3
9C3×S3
9C3×S3
18C3×S3
43- 1+2
4He3
43- 1+2
9S32
3C3×C3⋊S3
4C3≀C3

Character table of C33⋊A4

 class 123A3B3C3D3E3F69A9B9C9D
 size 1274461236365436363636
ρ11111111111111    trivial
ρ2111111ζ3ζ321ζ32ζ32ζ3ζ3    linear of order 3
ρ3111111ζ32ζ31ζ3ζ3ζ32ζ32    linear of order 3
ρ43-1333300-10000    orthogonal lifted from A4
ρ540-1-3-3/2-1+3-3/2-21ζ3ζ320ζ31ζ321    complex faithful
ρ640-1+3-3/2-1-3-3/2-21ζ32ζ30ζ321ζ31    complex faithful
ρ740-1+3-3/2-1-3-3/2-21ζ3ζ3201ζ31ζ32    complex faithful
ρ840-1-3-3/2-1+3-3/2-21110ζ32ζ3ζ3ζ32    complex faithful
ρ940-1+3-3/2-1-3-3/2-21110ζ3ζ32ζ32ζ3    complex faithful
ρ1040-1-3-3/2-1+3-3/2-21ζ32ζ301ζ321ζ3    complex faithful
ρ1162-3-33000-10000    orthogonal faithful
ρ126-2-3-3300010000    orthogonal faithful
ρ13120330-30000000    orthogonal faithful

Permutation representations of C33⋊A4
On 9 points - transitive group 9T25
Generators in S9
(7 8 9)
(4 6 5)
(1 3 2)
(5 6)(8 9)
(2 3)(5 6)
(1 7 4)(2 9 5)(3 8 6)

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

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

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

G:=TransitiveGroup(9,25);

On 12 points - transitive group 12T132
Generators in S12
(1 2 3)(4 5 6)(7 8 9)(10 11 12)
(1 2 3)(4 5 6)(7 9 8)(10 12 11)
(1 2 3)(4 6 5)(7 9 8)(10 11 12)
(1 4)(2 6)(3 5)(7 11)(8 10)(9 12)
(1 8)(2 9)(3 7)(4 10)(5 11)(6 12)
(1 3 2)(4 7 12)(5 9 10)(6 8 11)

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

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

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

G:=TransitiveGroup(12,132);

On 12 points - transitive group 12T133
Generators in S12
(1 2 3)(4 5 6)(7 8 9)(10 11 12)
(1 2 3)(4 5 6)(7 9 8)(10 12 11)
(1 2 3)(4 6 5)(7 9 8)(10 11 12)
(1 4)(2 6)(3 5)(7 11)(8 10)(9 12)
(1 8)(2 9)(3 7)(4 10)(5 11)(6 12)
(4 8 10)(5 7 11)(6 9 12)

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

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

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

G:=TransitiveGroup(12,133);

On 18 points - transitive group 18T141
Generators in S18
(13 14 15)(16 17 18)
(1 3 2)(4 5 7)
(6 10 11)(8 9 12)
(1 4)(2 5)(3 7)(6 12)(8 10)(9 11)(14 15)(16 17)
(2 3)(5 7)(6 12)(8 11)(9 10)(13 18)(14 16)(15 17)
(1 6 18)(2 11 17)(3 10 16)(4 12 13)(5 8 14)(7 9 15)

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

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

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

G:=TransitiveGroup(18,141);

On 18 points - transitive group 18T142
Generators in S18
(7 8 9)(10 11 12)(13 14 15)(16 17 18)
(1 3 2)(4 5 6)(7 8 9)(10 12 11)
(1 3 2)(4 6 5)(13 14 15)(16 18 17)
(1 4)(2 5)(3 6)(7 8)(11 12)(13 18)(14 17)(15 16)
(2 3)(5 6)(7 11)(8 12)(9 10)(13 16)(14 17)(15 18)
(1 14 9)(2 13 8)(3 15 7)(4 17 10)(5 16 11)(6 18 12)

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

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

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

G:=TransitiveGroup(18,142);

On 18 points - transitive group 18T143
Generators in S18
(13 14 15)(16 17 18)
(1 2 3)(4 5 6)
(7 9 8)(10 11 12)
(1 5)(2 4)(3 6)(13 16)(14 18)(15 17)
(1 5)(2 4)(3 6)(7 10)(8 11)(9 12)
(1 10 13)(2 11 14)(3 12 15)(4 8 18)(5 7 16)(6 9 17)

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

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

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

G:=TransitiveGroup(18,143);

On 27 points - transitive group 27T130
Generators in S27
(10 11 12)(13 14 15)(16 17 18)(19 20 21)(22 23 24)(25 26 27)
(1 7 4)(2 8 5)(3 9 6)(10 15 19)(11 13 20)(12 14 21)
(1 3 2)(4 6 5)(7 9 8)(16 27 22)(17 25 23)(18 26 24)
(4 7)(5 8)(6 9)(11 12)(13 21)(14 20)(15 19)(16 17)(22 23)(25 27)
(2 3)(4 7)(5 9)(6 8)(13 20)(14 21)(15 19)(22 27)(23 25)(24 26)
(1 18 10)(2 17 19)(3 16 15)(4 24 12)(5 23 21)(6 22 14)(7 26 11)(8 25 20)(9 27 13)

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

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

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

G:=TransitiveGroup(27,130);

On 27 points: primitive - transitive group 27T131
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 9)(10 20)(11 19)(12 21)(16 23)(17 22)(18 24)(25 27)
(4 15)(5 13)(6 14)(7 25)(8 26)(9 27)(10 23)(11 24)(12 22)(16 20)(17 21)(18 19)
(2 6 26)(3 14 8)(4 21 27)(5 12 9)(7 13 17)(10 23 20)(11 18 24)(15 22 25)

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

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

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

G:=TransitiveGroup(27,131);

Polynomial with Galois group C33⋊A4 over ℚ
actionf(x)Disc(f)
9T25x9+3x8-20x7-63x6+87x5+355x4+147x3-313x2-317x-8376·832·1812·42432
12T132x12-3x11-423x10+1162x9+63891x8-150879x7-4461428x6+9070833x5+148490544x4-264799046x3-2074843638x2+2972896206x+6882575909320·58·172·592·612·1638·4912
12T133x12-3x11-103x10+262x9+2631x8-8429x7-18568x6+92583x5-43306x4-228636x3+380622x2-213274x+35969312·58·292·1636·2272·2915119412075556472

Matrix representation of C33⋊A4 in GL4(𝔽7) generated by

1454
6340
3361
0002
,
3145
1335
0040
0002
,
6251
0405
4406
0002
,
2454
3615
1636
2253
,
1526
6656
3422
1135
,
4520
0062
0235
0002
G:=sub<GL(4,GF(7))| [1,6,3,0,4,3,3,0,5,4,6,0,4,0,1,2],[3,1,0,0,1,3,0,0,4,3,4,0,5,5,0,2],[6,0,4,0,2,4,4,0,5,0,0,0,1,5,6,2],[2,3,1,2,4,6,6,2,5,1,3,5,4,5,6,3],[1,6,3,1,5,6,4,1,2,5,2,3,6,6,2,5],[4,0,0,0,5,0,2,0,2,6,3,0,0,2,5,2] >;

C33⋊A4 in GAP, Magma, Sage, TeX

C_3^3\rtimes A_4
% in TeX

G:=Group("C3^3:A4");
// GroupNames label

G:=SmallGroup(324,160);
// by ID

G=gap.SmallGroup(324,160);
# by ID

G:=PCGroup([6,-3,-2,2,-3,3,3,109,56,867,111,3244,730,376,437,2603]);
// Polycyclic

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

Export

Subgroup lattice of C33⋊A4 in TeX
Character table of C33⋊A4 in TeX

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