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G = C3⋊Dic3  order 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, C32⋊3C4, 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 >

C1
C2
C3
C3
C3
C3
9C4
C6
C6
C6
C6
C32
3Dic3
3Dic3
3Dic3
3Dic3
C3×C6
C3⋊Dic3

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)]])
 

C3⋊Dic3 is a maximal subgroup of
 C32⋊2C8  S3×Dic3  D6⋊S3  C4×C3⋊S3  C32⋊7D4  C32⋊C12  C33⋊5C4  C6.5S4  C6.7S4  C32⋊3F5  C34⋊C4  C39⋊Dic3
 C3⋊Dic3p: C32⋊2Q8  C32⋊4Q8  C9⋊Dic3  C3⋊Dic15  C3⋊Dic21  C3⋊Dic33  C3⋊Dic39 ...
C3⋊Dic3 is a maximal quotient of
 He3⋊3C4  C33⋊5C4  C6.7S4  C32⋊3F5  C34⋊C4  C39⋊Dic3
 C6p.S3: C32⋊4C8  C9⋊Dic3  C3⋊Dic15  C3⋊Dic21  C3⋊Dic33  C3⋊Dic39 ...

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
 

Export

Subgroup lattice of C3⋊Dic3 in TeX
Character table of C3⋊Dic3 in TeX

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