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G = C33⋊C12  order 324 = 22·34

1st semidirect product of C33 and C12 acting via C12/C2=C6

metabelian, supersoluble, monomial

Aliases: C33⋊1C12, He3⋊1Dic3, C3≀C3⋊3C4, C33⋊5C4⋊1C3, (C2×He3).1S3, (C32×C6).2C6, C2.(C33⋊C6), C6.2(C32⋊C6), C3.2(C32⋊C12), C32.6(C3×Dic3), (C2×C3≀C3).3C2, (C3×C6).13(C3×S3), SmallGroup(324,14)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C33 — C33⋊C12
Chief series
C1 — C3 — C32 — C33 — C32×C6 — C2×C3≀C3 — C33⋊C12
Lower central
C33 — C33⋊C12
Upper central
C1 — C2

Generators and relations for C33⋊C12
 G = < a,b,c,d | a3=b3=c3=d12=1, ab=ba, ac=ca, dad-1=a-1bc-1, bc=cb, dbd-1=b-1c, dcd-1=c-1 >

C1
C2
C3
3C3
3C3
3C3
3C3
9C3
27C4
C6
3C6
3C6
3C6
3C6
9C6
C32
3C32
3C32
3C32
3C32
3C32
6C9
9Dic3
27Dic3
27C12
27Dic3
27Dic3
27Dic3
C3×C6
3C3×C6
3C3×C6
3C3×C6
3C3×C6
3C3×C6
6C18
He3
C33
23- 1+2
3C3⋊Dic3
9C3×Dic3
9C3⋊Dic3
9C3⋊Dic3
9C3⋊Dic3
9C3⋊Dic3
C2×He3
C32×C6
2C2×3- 1+2
C3≀C3
C33⋊5C4
3C32⋊C12
C2×C3≀C3
C33⋊C12

Character table of C33⋊C12

 class 123A3B3C3D3E3F3G4A4B6A6B6C6D6E6F6G9A9B12A12B12C12D18A18B
 size 112666699272726666991818272727271818
ρ111111111111111111111111111    trivial
ρ2111111111-1-1111111111-1-1-1-111    linear of order 2
ρ31111111ζ3ζ32-1-111111ζ32ζ3ζ3ζ32ζ6ζ65ζ65ζ6ζ32ζ3    linear of order 6
ρ41111111ζ32ζ3-1-111111ζ3ζ32ζ32ζ3ζ65ζ6ζ6ζ65ζ3ζ32    linear of order 6
ρ51111111ζ32ζ31111111ζ3ζ32ζ32ζ3ζ3ζ32ζ32ζ3ζ3ζ32    linear of order 3
ρ61111111ζ3ζ321111111ζ32ζ3ζ3ζ32ζ32ζ3ζ3ζ32ζ32ζ3    linear of order 3
ρ71-11111111-ii-1-1-1-1-1-1-111-ii-ii-1-1    linear of order 4
ρ81-11111111i-i-1-1-1-1-1-1-111i-ii-i-1-1    linear of order 4
ρ91-111111ζ3ζ32-ii-1-1-1-1-1ζ6ζ65ζ3ζ32ζ43ζ32ζ4ζ3ζ43ζ3ζ4ζ32ζ6ζ65    linear of order 12
ρ101-111111ζ3ζ32i-i-1-1-1-1-1ζ6ζ65ζ3ζ32ζ4ζ32ζ43ζ3ζ4ζ3ζ43ζ32ζ6ζ65    linear of order 12
ρ111-111111ζ32ζ3i-i-1-1-1-1-1ζ65ζ6ζ32ζ3ζ4ζ3ζ43ζ32ζ4ζ32ζ43ζ3ζ65ζ6    linear of order 12
ρ121-111111ζ32ζ3-ii-1-1-1-1-1ζ65ζ6ζ32ζ3ζ43ζ3ζ4ζ32ζ43ζ32ζ4ζ3ζ65ζ6    linear of order 12
ρ13222-1-12-122002-12-1-122-1-10000-1-1    orthogonal lifted from S3
ρ142-22-1-12-12200-21-211-2-2-1-1000011    symplectic lifted from Dic3, Schur index 2
ρ15222-1-12-1-1+√-3-1-√-3002-12-1-1-1-√-3-1+√-3ζ65ζ60000ζ6ζ65    complex lifted from C3×S3
ρ162-22-1-12-1-1+√-3-1-√-300-21-2111+√-31-√-3ζ65ζ60000ζ32ζ3    complex lifted from C3×Dic3
ρ172-22-1-12-1-1-√-3-1+√-300-21-2111-√-31+√-3ζ6ζ650000ζ3ζ32    complex lifted from C3×Dic3
ρ18222-1-12-1-1-√-3-1+√-3002-12-1-1-1+√-3-1-√-3ζ6ζ650000ζ65ζ6    complex lifted from C3×S3
ρ1966-3-30030000-3003-30000000000    orthogonal lifted from C33⋊C6
ρ2066-3030-30000-330-300000000000    orthogonal lifted from C33⋊C6
ρ2166-33-3000000-3-30030000000000    orthogonal lifted from C33⋊C6
ρ2266600-30000060-3000000000000    orthogonal lifted from C32⋊C6
ρ236-6-3-30030000300-330000000000    symplectic faithful, Schur index 2
ρ246-6-3030-300003-30300000000000    symplectic faithful, Schur index 2
ρ256-6600-300000-603000000000000    symplectic lifted from C32⋊C12, Schur index 2
ρ266-6-33-30000003300-30000000000    symplectic faithful, Schur index 2

Smallest permutation representation of C33⋊C12
►On 36 points
Generators in S36
(2 33 19)(5 22 36)(8 27 13)(11 16 30)

(2 19 33)(3 20 34)(5 36 22)(6 25 23)(8 13 27)(9 14 28)(11 30 16)(12 31 17)

(1 18 32)(2 33 19)(3 20 34)(4 35 21)(5 22 36)(6 25 23)(7 24 26)(8 27 13)(9 14 28)(10 29 15)(11 16 30)(12 31 17)

(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 28 29 30 31 32 33 34 35 36)
 
G:=sub<Sym(36)| (2,33,19)(5,22,36)(8,27,13)(11,16,30), (2,19,33)(3,20,34)(5,36,22)(6,25,23)(8,13,27)(9,14,28)(11,30,16)(12,31,17), (1,18,32)(2,33,19)(3,20,34)(4,35,21)(5,22,36)(6,25,23)(7,24,26)(8,27,13)(9,14,28)(10,29,15)(11,16,30)(12,31,17), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36)>;
 
G:=Group( (2,33,19)(5,22,36)(8,27,13)(11,16,30), (2,19,33)(3,20,34)(5,36,22)(6,25,23)(8,13,27)(9,14,28)(11,30,16)(12,31,17), (1,18,32)(2,33,19)(3,20,34)(4,35,21)(5,22,36)(6,25,23)(7,24,26)(8,27,13)(9,14,28)(10,29,15)(11,16,30)(12,31,17), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36) );
 
G=PermutationGroup([[(2,33,19),(5,22,36),(8,27,13),(11,16,30)], [(2,19,33),(3,20,34),(5,36,22),(6,25,23),(8,13,27),(9,14,28),(11,30,16),(12,31,17)], [(1,18,32),(2,33,19),(3,20,34),(4,35,21),(5,22,36),(6,25,23),(7,24,26),(8,27,13),(9,14,28),(10,29,15),(11,16,30),(12,31,17)], [(1,2,3,4,5,6,7,8,9,10,11,12),(13,14,15,16,17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32,33,34,35,36)]])
 

Matrix representation of C33⋊C12 ►in GL8(𝔽37)

3636000000
10000000
00100000
00010000
00001000
00000100
00000001
0000003636
,
10000000
01000000
00100000
00010000
00000100
0000363600
0000003636
00000010
,
10000000
01000000
00010000
0036360000
00000100
0000363600
00000001
0000003636
,
60000000
3131000000
000051000
000053200
000000510
000000532
005100000
005320000

G:=sub<GL(8,GF(37))| [36,1,0,0,0,0,0,0,36,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,36,0,0,0,0,0,0,1,36],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,36,0,0,0,0,0,0,1,36,0,0,0,0,0,0,0,0,36,1,0,0,0,0,0,0,36,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,36,0,0,0,0,0,0,1,36,0,0,0,0,0,0,0,0,0,36,0,0,0,0,0,0,1,36,0,0,0,0,0,0,0,0,0,36,0,0,0,0,0,0,1,36],[6,31,0,0,0,0,0,0,0,31,0,0,0,0,0,0,0,0,0,0,0,0,5,5,0,0,0,0,0,0,10,32,0,0,5,5,0,0,0,0,0,0,10,32,0,0,0,0,0,0,0,0,5,5,0,0,0,0,0,0,10,32,0,0] >;
 

C33⋊C12 in GAP, Magma, Sage, TeX 

C_3^3\rtimes C_{12}
 
% in TeX
 
G:=Group("C3^3:C12");
 
// GroupNames label
 
G:=SmallGroup(324,14);
 
// by ID
 
G=gap.SmallGroup(324,14);
 
# by ID
 
G:=PCGroup([6,-2,-3,-2,-3,-3,-3,36,579,585,2164,2170,7781]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^3=b^3=c^3=d^12=1,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1*b*c^-1,b*c=c*b,d*b*d^-1=b^-1*c,d*c*d^-1=c^-1>;
 
// generators/relations
 

Export

Subgroup lattice of C33⋊C12 in TeX
Character table of C33⋊C12 in TeX

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