Copied to
clipboard

## G = C3⋊Dic3order 36 = 22·32

### The semidirect product of C3 and Dic3 acting via Dic3/C6=C2

Aliases: C3⋊Dic3, C6.3S3, C323C4, C2.(C3⋊S3), (C3×C6).2C2, SmallGroup(36,7)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C3⋊Dic3
 Chief series C1 — C3 — C32 — C3×C6 — C3⋊Dic3
 Lower central C32 — C3⋊Dic3
 Upper central C1 — C2

Generators and relations for C3⋊Dic3
G = < a,b,c | a3=b6=1, c2=b3, ab=ba, cac-1=a-1, cbc-1=b-1 >

Character table of C3⋊Dic3

 class 1 2 3A 3B 3C 3D 4A 4B 6A 6B 6C 6D size 1 1 2 2 2 2 9 9 2 2 2 2 ρ1 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 linear of order 2 ρ3 1 -1 1 1 1 1 -i i -1 -1 -1 -1 linear of order 4 ρ4 1 -1 1 1 1 1 i -i -1 -1 -1 -1 linear of order 4 ρ5 2 2 -1 -1 2 -1 0 0 -1 2 -1 -1 orthogonal lifted from S3 ρ6 2 2 2 -1 -1 -1 0 0 -1 -1 -1 2 orthogonal lifted from S3 ρ7 2 2 -1 -1 -1 2 0 0 -1 -1 2 -1 orthogonal lifted from S3 ρ8 2 2 -1 2 -1 -1 0 0 2 -1 -1 -1 orthogonal lifted from S3 ρ9 2 -2 -1 -1 2 -1 0 0 1 -2 1 1 symplectic lifted from Dic3, Schur index 2 ρ10 2 -2 2 -1 -1 -1 0 0 1 1 1 -2 symplectic lifted from Dic3, Schur index 2 ρ11 2 -2 -1 2 -1 -1 0 0 -2 1 1 1 symplectic lifted from Dic3, Schur index 2 ρ12 2 -2 -1 -1 -1 2 0 0 1 1 -2 1 symplectic lifted from Dic3, Schur index 2

Smallest permutation representation of C3⋊Dic3
Regular action on 36 points
Generators in S36
(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)(2,21,30)(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)(2,21,30)(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),(2,21,30),(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 C3⋊Dic3 in GL4(𝔽13) generated by

 1 0 0 0 0 1 0 0 0 0 12 1 0 0 12 0
,
 0 12 0 0 1 1 0 0 0 0 1 0 0 0 0 1
,
 4 2 0 0 11 9 0 0 0 0 12 1 0 0 0 1
G:=sub<GL(4,GF(13))| [1,0,0,0,0,1,0,0,0,0,12,12,0,0,1,0],[0,1,0,0,12,1,0,0,0,0,1,0,0,0,0,1],[4,11,0,0,2,9,0,0,0,0,12,0,0,0,1,1] >;

C3⋊Dic3 in GAP, Magma, Sage, TeX

C_3\rtimes {\rm Dic}_3
% in TeX

G:=Group("C3:Dic3");
// GroupNames label

G:=SmallGroup(36,7);
// by ID

G=gap.SmallGroup(36,7);
# by ID

G:=PCGroup([4,-2,-2,-3,-3,8,98,387]);
// Polycyclic

G:=Group<a,b,c|a^3=b^6=1,c^2=b^3,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=b^-1>;
// generators/relations

׿
×
𝔽