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G = C33⋊Dic3  order 324 = 22·34

2nd semidirect product of C33 and Dic3 acting via Dic3/C2=S3

non-abelian, supersoluble, monomial

Aliases: He3⋊2Dic3, C33⋊2Dic3, 3- 1+2⋊1Dic3, C3≀C3⋊1C4, (C2×He3).4S3, (C32×C6).4S3, C2.(C33⋊S3), C6.2(He3⋊C2), C3.2(He3⋊3C4), C32.1(C3⋊Dic3), (C2×3- 1+2).1S3, (C2×C3≀C3).1C2, (C3×C6).1(C3⋊S3), SmallGroup(324,22)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C32 — C3≀C3 — C33⋊Dic3
Chief series
C1 — C3 — C32 — C33 — C3≀C3 — C2×C3≀C3 — C33⋊Dic3
Lower central
C3≀C3 — C33⋊Dic3
Upper central
C1 — C2

Generators and relations for C33⋊Dic3
 G = < a,b,c,d,e | a3=b3=c3=d6=1, e2=d3, ab=ba, ac=ca, dad-1=ab-1c, eae-1=a-1, bc=cb, dbd-1=ebe-1=bc-1, cd=dc, ece-1=c-1, ede-1=d-1 >

Subgroups: 278 in 60 conjugacy classes, 17 normal (13 characteristic)
C1, C2, C3, C3, C4, C6, C6, C9, C32, C32, Dic3, C12, C18, C3×C6, C3×C6, He3, 3- 1+2, C33, Dic9, C3×Dic3, C3⋊Dic3, C2×He3, C2×3- 1+2, C32×C6, C3≀C3, C32⋊C12, C9⋊C12, C3×C3⋊Dic3, C2×C3≀C3, C33⋊Dic3
Quotients: C1, C2, C4, S3, Dic3, C3⋊S3, C3⋊Dic3, He3⋊C2, He3⋊3C4, C33⋊S3, C33⋊Dic3

Character table of C33⋊Dic3

 class 123A3B3C3D3E3F3G4A4B6A6B6C6D6E6F6G9A9B12A12B12C12D18A18B
 size 11233666182727233666181818272727271818
ρ111111111111111111111111111    trivial
ρ2111111111-1-1111111111-1-1-1-111    linear of order 2
ρ31-11111111i-i-1-1-1-1-1-1-111i-i-ii-1-1    linear of order 4
ρ41-11111111-ii-1-1-1-1-1-1-111-iii-i-1-1    linear of order 4
ρ522222-1-1-1-100222-1-1-1-12-10000-12    orthogonal lifted from S3
ρ622222222-100222222-1-1-10000-1-1    orthogonal lifted from S3
ρ722222-1-1-1200222-1-1-12-1-10000-1-1    orthogonal lifted from S3
ρ822222-1-1-1-100222-1-1-1-1-1200002-1    orthogonal lifted from S3
ρ92-2222-1-1-1-100-2-2-211112-100001-2    symplectic lifted from Dic3, Schur index 2
ρ102-2222-1-1-1-100-2-2-21111-120000-21    symplectic lifted from Dic3, Schur index 2
ρ112-2222-1-1-1200-2-2-2111-2-1-1000011    symplectic lifted from Dic3, Schur index 2
ρ122-2222222-100-2-2-2-2-2-21-1-1000011    symplectic lifted from Dic3, Schur index 2
ρ13333-3+3√-3/2-3-3√-3/20000-1-13-3-3√-3/2-3+3√-3/2000000ζ6ζ65ζ6ζ6500    complex lifted from He3⋊C2
ρ14333-3+3√-3/2-3-3√-3/20000113-3-3√-3/2-3+3√-3/2000000ζ32ζ3ζ32ζ300    complex lifted from He3⋊C2
ρ15333-3-3√-3/2-3+3√-3/20000113-3+3√-3/2-3-3√-3/2000000ζ3ζ32ζ3ζ3200    complex lifted from He3⋊C2
ρ16333-3-3√-3/2-3+3√-3/20000-1-13-3+3√-3/2-3-3√-3/2000000ζ65ζ6ζ65ζ600    complex lifted from He3⋊C2
ρ173-33-3+3√-3/2-3-3√-3/20000i-i-33+3√-3/23-3√-3/2000000ζ4ζ32ζ43ζ3ζ43ζ32ζ4ζ300    complex lifted from He3⋊3C4
ρ183-33-3+3√-3/2-3-3√-3/20000-ii-33+3√-3/23-3√-3/2000000ζ43ζ32ζ4ζ3ζ4ζ32ζ43ζ300    complex lifted from He3⋊3C4
ρ193-33-3-3√-3/2-3+3√-3/20000-ii-33-3√-3/23+3√-3/2000000ζ43ζ3ζ4ζ32ζ4ζ3ζ43ζ3200    complex lifted from He3⋊3C4
ρ203-33-3-3√-3/2-3+3√-3/20000i-i-33-3√-3/23+3√-3/2000000ζ4ζ3ζ43ζ32ζ43ζ3ζ4ζ3200    complex lifted from He3⋊3C4
ρ2166-30003-3000-300-303000000000    orthogonal lifted from C33⋊S3
ρ2266-3003-30000-30003-3000000000    orthogonal lifted from C33⋊S3
ρ2366-300-303000-3003-30000000000    orthogonal lifted from C33⋊S3
ρ246-6-30003-300030030-3000000000    symplectic faithful, Schur index 2
ρ256-6-300-303000300-330000000000    symplectic faithful, Schur index 2
ρ266-6-3003-300003000-33000000000    symplectic faithful, Schur index 2

Smallest permutation representation of C33⋊Dic3
►On 36 points
Generators in S36
(1 20 29)(2 30 21)(3 22 25)(4 23 26)(5 27 24)(6 19 28)(7 31 14)(8 32 15)(9 16 33)(10 34 17)(11 35 18)(12 13 36)

(2 30 21)(3 22 25)(5 27 24)(6 19 28)(7 31 14)(9 16 33)(10 34 17)(12 13 36)

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

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

Matrix representation of C33⋊Dic3 ►in GL9(𝔽37)

362536000000
010000000
100000000
000010000
00036360000
000000100
00000363600
00000003636
000000010
,
1000000000
0100000000
0010000000
000100000
000010000
000000100
00000363600
00000003636
000000010
,
100000000
010000000
001000000
000010000
00036360000
000000100
00000363600
000000001
00000003636
,
3600000000
11111000000
0027000000
000001000
000000100
000000010
000000001
000100000
000010000
,
3100000000
0310000000
6356000000
000001000
00000363600
000100000
00036360000
000000010
00000003636

G:=sub<GL(9,GF(37))| [36,0,1,0,0,0,0,0,0,25,1,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,1,36,0,0,0,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,1,36,0,0,0,0,0,0,0,0,0,36,1,0,0,0,0,0,0,0,36,0],[10,0,0,0,0,0,0,0,0,0,10,0,0,0,0,0,0,0,0,0,10,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,1,36,0,0,0,0,0,0,0,0,0,36,1,0,0,0,0,0,0,0,36,0],[1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,1,36,0,0,0,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,1,36,0,0,0,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,1,36],[36,1,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,11,27,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0],[31,0,6,0,0,0,0,0,0,0,31,35,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,0,0,1,36,0,0,0,0,0,0,0,0,36,0,0,0,0,0,1,36,0,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,0,0,0,0,1,36,0,0,0,0,0,0,0,0,36] >;
 

C33⋊Dic3 in GAP, Magma, Sage, TeX 

C_3^3\rtimes {\rm Dic}_3
 
% in TeX
 
G:=Group("C3^3:Dic3");
 
// GroupNames label
 
G:=SmallGroup(324,22);
 
// by ID
 
G=gap.SmallGroup(324,22);
 
# by ID
 
G:=PCGroup([6,-2,-2,-3,-3,-3,-3,12,146,579,303,7564,1096,7781]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^6=1,e^2=d^3,a*b=b*a,a*c=c*a,d*a*d^-1=a*b^-1*c,e*a*e^-1=a^-1,b*c=c*b,d*b*d^-1=e*b*e^-1=b*c^-1,c*d=d*c,e*c*e^-1=c^-1,e*d*e^-1=d^-1>;
 
// generators/relations
 

Export

Character table of C33⋊Dic3 in TeX

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