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G = C3⋊Dic3order 36 = 22·32

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

metabelian, supersoluble, monomial, A-group

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

Series: Derived Chief Lower central Upper central

C1C32 — C3⋊Dic3
C1C3C32C3×C6 — C3⋊Dic3
C32 — C3⋊Dic3
C1C2

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 >

9C4
3Dic3
3Dic3
3Dic3
3Dic3

Character table of C3⋊Dic3

 class 123A3B3C3D4A4B6A6B6C6D
 size 112222992222
ρ1111111111111    trivial
ρ2111111-1-11111    linear of order 2
ρ31-11111-ii-1-1-1-1    linear of order 4
ρ41-11111i-i-1-1-1-1    linear of order 4
ρ522-1-12-100-12-1-1    orthogonal lifted from S3
ρ6222-1-1-100-1-1-12    orthogonal lifted from S3
ρ722-1-1-1200-1-12-1    orthogonal lifted from S3
ρ822-12-1-1002-1-1-1    orthogonal lifted from S3
ρ92-2-1-12-1001-211    symplectic lifted from Dic3, Schur index 2
ρ102-22-1-1-100111-2    symplectic lifted from Dic3, Schur index 2
ρ112-2-12-1-100-2111    symplectic lifted from Dic3, Schur index 2
ρ122-2-1-1-120011-21    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

1000
0100
00121
00120
,
01200
1100
0010
0001
,
4200
11900
00121
0001
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

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