Copied to
clipboard

## G = C33⋊Dic3order 324 = 22·34

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

Aliases: He32Dic3, C332Dic3, 3- 1+21Dic3, C3≀C31C4, (C2×He3).4S3, (C32×C6).4S3, C2.(C33⋊S3), C6.2(He3⋊C2), C3.2(He33C4), 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, He33C4, C33⋊S3, C33⋊Dic3

Character table of C33⋊Dic3

 class 1 2 3A 3B 3C 3D 3E 3F 3G 4A 4B 6A 6B 6C 6D 6E 6F 6G 9A 9B 12A 12B 12C 12D 18A 18B size 1 1 2 3 3 6 6 6 18 27 27 2 3 3 6 6 6 18 18 18 27 27 27 27 18 18 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 1 1 1 1 1 -1 -1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 1 1 linear of order 2 ρ3 1 -1 1 1 1 1 1 1 1 i -i -1 -1 -1 -1 -1 -1 -1 1 1 i -i -i i -1 -1 linear of order 4 ρ4 1 -1 1 1 1 1 1 1 1 -i i -1 -1 -1 -1 -1 -1 -1 1 1 -i i i -i -1 -1 linear of order 4 ρ5 2 2 2 2 2 -1 -1 -1 -1 0 0 2 2 2 -1 -1 -1 -1 2 -1 0 0 0 0 -1 2 orthogonal lifted from S3 ρ6 2 2 2 2 2 2 2 2 -1 0 0 2 2 2 2 2 2 -1 -1 -1 0 0 0 0 -1 -1 orthogonal lifted from S3 ρ7 2 2 2 2 2 -1 -1 -1 2 0 0 2 2 2 -1 -1 -1 2 -1 -1 0 0 0 0 -1 -1 orthogonal lifted from S3 ρ8 2 2 2 2 2 -1 -1 -1 -1 0 0 2 2 2 -1 -1 -1 -1 -1 2 0 0 0 0 2 -1 orthogonal lifted from S3 ρ9 2 -2 2 2 2 -1 -1 -1 -1 0 0 -2 -2 -2 1 1 1 1 2 -1 0 0 0 0 1 -2 symplectic lifted from Dic3, Schur index 2 ρ10 2 -2 2 2 2 -1 -1 -1 -1 0 0 -2 -2 -2 1 1 1 1 -1 2 0 0 0 0 -2 1 symplectic lifted from Dic3, Schur index 2 ρ11 2 -2 2 2 2 -1 -1 -1 2 0 0 -2 -2 -2 1 1 1 -2 -1 -1 0 0 0 0 1 1 symplectic lifted from Dic3, Schur index 2 ρ12 2 -2 2 2 2 2 2 2 -1 0 0 -2 -2 -2 -2 -2 -2 1 -1 -1 0 0 0 0 1 1 symplectic lifted from Dic3, Schur index 2 ρ13 3 3 3 -3+3√-3/2 -3-3√-3/2 0 0 0 0 -1 -1 3 -3-3√-3/2 -3+3√-3/2 0 0 0 0 0 0 ζ6 ζ65 ζ6 ζ65 0 0 complex lifted from He3⋊C2 ρ14 3 3 3 -3+3√-3/2 -3-3√-3/2 0 0 0 0 1 1 3 -3-3√-3/2 -3+3√-3/2 0 0 0 0 0 0 ζ32 ζ3 ζ32 ζ3 0 0 complex lifted from He3⋊C2 ρ15 3 3 3 -3-3√-3/2 -3+3√-3/2 0 0 0 0 1 1 3 -3+3√-3/2 -3-3√-3/2 0 0 0 0 0 0 ζ3 ζ32 ζ3 ζ32 0 0 complex lifted from He3⋊C2 ρ16 3 3 3 -3-3√-3/2 -3+3√-3/2 0 0 0 0 -1 -1 3 -3+3√-3/2 -3-3√-3/2 0 0 0 0 0 0 ζ65 ζ6 ζ65 ζ6 0 0 complex lifted from He3⋊C2 ρ17 3 -3 3 -3+3√-3/2 -3-3√-3/2 0 0 0 0 i -i -3 3+3√-3/2 3-3√-3/2 0 0 0 0 0 0 ζ4ζ32 ζ43ζ3 ζ43ζ32 ζ4ζ3 0 0 complex lifted from He3⋊3C4 ρ18 3 -3 3 -3+3√-3/2 -3-3√-3/2 0 0 0 0 -i i -3 3+3√-3/2 3-3√-3/2 0 0 0 0 0 0 ζ43ζ32 ζ4ζ3 ζ4ζ32 ζ43ζ3 0 0 complex lifted from He3⋊3C4 ρ19 3 -3 3 -3-3√-3/2 -3+3√-3/2 0 0 0 0 -i i -3 3-3√-3/2 3+3√-3/2 0 0 0 0 0 0 ζ43ζ3 ζ4ζ32 ζ4ζ3 ζ43ζ32 0 0 complex lifted from He3⋊3C4 ρ20 3 -3 3 -3-3√-3/2 -3+3√-3/2 0 0 0 0 i -i -3 3-3√-3/2 3+3√-3/2 0 0 0 0 0 0 ζ4ζ3 ζ43ζ32 ζ43ζ3 ζ4ζ32 0 0 complex lifted from He3⋊3C4 ρ21 6 6 -3 0 0 0 3 -3 0 0 0 -3 0 0 -3 0 3 0 0 0 0 0 0 0 0 0 orthogonal lifted from C33⋊S3 ρ22 6 6 -3 0 0 3 -3 0 0 0 0 -3 0 0 0 3 -3 0 0 0 0 0 0 0 0 0 orthogonal lifted from C33⋊S3 ρ23 6 6 -3 0 0 -3 0 3 0 0 0 -3 0 0 3 -3 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C33⋊S3 ρ24 6 -6 -3 0 0 0 3 -3 0 0 0 3 0 0 3 0 -3 0 0 0 0 0 0 0 0 0 symplectic faithful, Schur index 2 ρ25 6 -6 -3 0 0 -3 0 3 0 0 0 3 0 0 -3 3 0 0 0 0 0 0 0 0 0 0 symplectic faithful, Schur index 2 ρ26 6 -6 -3 0 0 3 -3 0 0 0 0 3 0 0 0 -3 3 0 0 0 0 0 0 0 0 0 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)

 36 25 36 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 36 36 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 36 36 0 0 0 0 0 0 0 0 0 36 36 0 0 0 0 0 0 0 1 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 1 0 0 0 0 0 0 0 36 36 0 0 0 0 0 0 0 0 0 36 36 0 0 0 0 0 0 0 1 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 1 0 0 0 0 0 0 0 36 36 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 36 36 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 36 36
,
 36 0 0 0 0 0 0 0 0 1 11 11 0 0 0 0 0 0 0 0 27 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 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
,
 31 0 0 0 0 0 0 0 0 0 31 0 0 0 0 0 0 0 6 35 6 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 36 36 0 0 0 0 0 1 0 0 0 0 0 0 0 0 36 36 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 36 36

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

׿
×
𝔽