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G = C33⋊D4order 216 = 23·33

2nd semidirect product of C33 and D4 acting faithfully

non-abelian, soluble, monomial

Aliases: C332D4, S32⋊S3, C32S3≀C2, C3⋊S3.1D6, C33⋊C42C2, C324D61C2, C323(C3⋊D4), (C3×S32)⋊2C2, (C3×C3⋊S3).4C22, SmallGroup(216,158)

Series: Derived Chief Lower central Upper central

C1C32C3×C3⋊S3 — C33⋊D4
C1C3C33C3×C3⋊S3C324D6 — C33⋊D4
C33C3×C3⋊S3 — C33⋊D4
C1

Generators and relations for C33⋊D4
 G = < a,b,c,d,e | a3=b3=c3=d4=e2=1, ab=ba, ac=ca, dad-1=b-1, eae=b, bc=cb, dbd-1=ebe=a, dcd-1=ece=c-1, ede=d-1 >

Subgroups: 384 in 60 conjugacy classes, 11 normal (all characteristic)
C1, C2 [×3], C3, C3 [×4], C4, C22 [×2], S3 [×6], C6 [×5], D4, C32, C32 [×4], Dic3, D6 [×3], C2×C6, C3×S3 [×8], C3⋊S3, C3⋊S3, C3×C6, C3⋊D4, C33, C32⋊C4, S32, S32 [×2], S3×C6, S3×C32, C3×C3⋊S3, C3×C3⋊S3, S3≀C2, C33⋊C4, C3×S32, C324D6, C33⋊D4
Quotients: C1, C2 [×3], C22, S3, D4, D6, C3⋊D4, S3≀C2, C33⋊D4

Character table of C33⋊D4

 class 12A2B2C3A3B3C3D3E3F46A6B6C6D6E6F6G
 size 1691824444854661212121836
ρ1111111111111111111    trivial
ρ2111-1111111-1111111-1    linear of order 2
ρ31-11-11111111-1-1-1-1-11-1    linear of order 2
ρ41-111111111-1-1-1-1-1-111    linear of order 2
ρ520-20222222000000-20    orthogonal lifted from D4
ρ62-220-12-1-12-101111-2-10    orthogonal lifted from D6
ρ72220-12-1-12-10-1-1-1-12-10    orthogonal lifted from S3
ρ820-20-12-1-12-10--3-3-3--3010    complex lifted from C3⋊D4
ρ920-20-12-1-12-10-3--3--3-3010    complex lifted from C3⋊D4
ρ1042004111-2-2022-1-1-100    orthogonal lifted from S3≀C2
ρ114-2004111-2-20-2-211100    orthogonal lifted from S3≀C2
ρ12400-24-2-2-21100000001    orthogonal lifted from S3≀C2
ρ1340024-2-2-2110000000-1    orthogonal lifted from S3≀C2
ρ144200-21-1-3-3/2-1+3-3/2-210-1+-3-1--3ζ6ζ65-100    complex faithful
ρ154-200-21-1+3-3/2-1-3-3/2-2101+-31--3ζ3ζ32100    complex faithful
ρ164200-21-1+3-3/2-1-3-3/2-210-1--3-1+-3ζ65ζ6-100    complex faithful
ρ174-200-21-1-3-3/2-1+3-3/2-2101--31+-3ζ32ζ3100    complex faithful
ρ188000-4-4222-100000000    orthogonal faithful

Permutation representations of C33⋊D4
On 12 points - transitive group 12T116
Generators in S12
(2 10 5)(4 7 12)
(1 8 9)(3 11 6)
(1 9 8)(2 5 10)(3 11 6)(4 7 12)
(1 2 3 4)(5 6 7 8)(9 10 11 12)
(1 4)(2 3)(5 6)(7 8)(9 12)(10 11)

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

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

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

G:=TransitiveGroup(12,116);

On 12 points - transitive group 12T120
Generators in S12
(1 9 8)(2 10 5)(3 6 11)(4 7 12)
(1 8 9)(2 10 5)(3 11 6)(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)
(2 4)(5 12)(6 11)(7 10)(8 9)

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

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

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

G:=TransitiveGroup(12,120);

On 18 points - transitive group 18T103
Generators in S18
(1 9 7)(3 11 13)(6 16 18)
(2 10 8)(4 12 14)(5 17 15)
(1 6 3)(2 4 5)(7 18 13)(8 14 15)(9 16 11)(10 12 17)
(1 2)(3 4)(5 6)(7 8 9 10)(11 12 13 14)(15 16 17 18)
(1 2)(3 4)(5 6)(7 8)(9 10)(11 12)(13 14)(15 18)(16 17)

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

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

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

G:=TransitiveGroup(18,103);

On 18 points - transitive group 18T106
Generators in S18
(1 13 11)(2 14 12)(3 7 9)(4 10 8)(5 16 18)(6 17 15)
(1 11 13)(2 14 12)(3 7 9)(4 8 10)(5 18 16)(6 17 15)
(1 5 4)(2 3 6)(7 17 14)(8 11 18)(9 15 12)(10 13 16)
(1 2)(3 4)(5 6)(7 8 9 10)(11 12 13 14)(15 16 17 18)
(3 6)(4 5)(7 17)(8 16)(9 15)(10 18)(11 13)

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

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

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

G:=TransitiveGroup(18,106);

On 24 points - transitive group 24T556
Generators in S24
(1 20 21)(3 23 18)(5 14 10)(7 12 16)
(2 17 22)(4 24 19)(6 15 11)(8 9 13)
(1 21 20)(2 17 22)(3 23 18)(4 19 24)(5 14 10)(6 11 15)(7 16 12)(8 9 13)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)
(1 8)(2 7)(3 6)(4 5)(9 20)(10 19)(11 18)(12 17)(13 21)(14 24)(15 23)(16 22)

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

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

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

G:=TransitiveGroup(24,556);

On 24 points - transitive group 24T560
Generators in S24
(1 20 21)(2 22 17)(3 23 18)(4 19 24)(5 14 10)(6 15 11)(7 12 16)(8 9 13)
(1 20 21)(2 17 22)(3 23 18)(4 24 19)(5 10 14)(6 15 11)(7 16 12)(8 9 13)
(1 21 20)(2 17 22)(3 23 18)(4 19 24)(5 14 10)(6 11 15)(7 16 12)(8 9 13)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)
(1 8)(2 7)(3 6)(4 5)(9 20)(10 19)(11 18)(12 17)(13 21)(14 24)(15 23)(16 22)

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

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

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

G:=TransitiveGroup(24,560);

On 27 points - transitive group 27T81
Generators in S27
(1 6 4)(2 27 25)(3 21 23)(5 18 17)(7 19 16)(8 12 24)(9 22 15)(10 26 14)(11 13 20)
(1 7 5)(2 22 20)(3 24 26)(4 16 17)(6 19 18)(8 14 21)(9 13 25)(10 23 12)(11 27 15)
(1 3 2)(4 23 25)(5 26 20)(6 21 27)(7 24 22)(8 15 19)(9 16 12)(10 13 17)(11 18 14)
(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)
(2 3)(4 5)(6 7)(8 15)(9 14)(10 13)(11 12)(16 18)(20 23)(21 22)(24 27)(25 26)

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

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

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

G:=TransitiveGroup(27,81);

C33⋊D4 is a maximal subgroup of   S3×S3≀C2
C33⋊D4 is a maximal quotient of   C3⋊S3.2D12  S32⋊Dic3  C33⋊C4⋊C4  C33⋊D8  C336SD16  C337SD16  C33⋊Q16

Polynomial with Galois group C33⋊D4 over ℚ
actionf(x)Disc(f)
12T116x12-3x9+3x6-3x3+3323·73
12T120x12-2x9-12x6-8x3-2-220·321·76

Matrix representation of C33⋊D4 in GL4(𝔽7) generated by

3243
4556
3361
0001
,
6211
2661
0010
0002
,
3145
1335
0040
0002
,
0515
1521
6146
1135
,
6330
4462
3425
6642
G:=sub<GL(4,GF(7))| [3,4,3,0,2,5,3,0,4,5,6,0,3,6,1,1],[6,2,0,0,2,6,0,0,1,6,1,0,1,1,0,2],[3,1,0,0,1,3,0,0,4,3,4,0,5,5,0,2],[0,1,6,1,5,5,1,1,1,2,4,3,5,1,6,5],[6,4,3,6,3,4,4,6,3,6,2,4,0,2,5,2] >;

C33⋊D4 in GAP, Magma, Sage, TeX

C_3^3\rtimes D_4
% in TeX

G:=Group("C3^3:D4");
// GroupNames label

G:=SmallGroup(216,158);
// by ID

G=gap.SmallGroup(216,158);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,3,-3,73,579,201,111,244,376,5189]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^4=e^2=1,a*b=b*a,a*c=c*a,d*a*d^-1=b^-1,e*a*e=b,b*c=c*b,d*b*d^-1=e*b*e=a,d*c*d^-1=e*c*e=c^-1,e*d*e=d^-1>;
// generators/relations

Export

Character table of C33⋊D4 in TeX

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