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G = He3⋊4Dic3  order 324 = 22·34

The semidirect product of He3 and Dic3 acting via Dic3/C3=C4

non-abelian, soluble

Aliases: He3⋊4Dic3, C3⋊(He3⋊C4), (C3×He3)⋊2C4, He3⋊C2.2S3, C3.2(C33⋊C4), C32.2(C32⋊C4), (C3×He3⋊C2).2C2, SmallGroup(324,113)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3 — C3×He3 — He3⋊4Dic3
Chief series
C1 — C3 — C32 — C3×He3 — C3×He3⋊C2 — He3⋊4Dic3
Lower central
C3×He3 — He3⋊4Dic3
Upper central
C1 — C3

Generators and relations for He3⋊4Dic3
 G = < a,b,c,d,e | a3=b3=c3=d6=1, e2=d3, ab=ba, cac-1=ab-1, dad-1=a-1b, eae-1=a-1bc-1, bc=cb, bd=db, be=eb, dcd-1=bc-1, ece-1=a-1bc, ede-1=d-1 >

C1
9C2
C3
C3
2C3
6C3
6C3
12C3
12C3
27C4
6S3
6S3
9C6
9C6
18C6
C32
2C32
2C32
4C32
4C32
6C32
6C32
6C32
6C32
6C32
6C32
9Dic3
27C12
6C3×S3
6C3×S3
6C3×S3
6C3×S3
6C3×S3
6C3×S3
6C3×S3
6C3×S3
9C3×C6
He3
2C33
2C33
4He3
4He3
9C3×Dic3
He3⋊C2
6S3×C32
6S3×C32
C3×He3
3He3⋊C4
C3×He3⋊C2
He3⋊4Dic3

Character table of He3⋊4Dic3

 class 123A3B3C3D3E3F3G3H3I3J3K4A4B6A6B6C6D6E12A12B12C12D
 size 191122212121212121227279918181827272727
ρ1111111111111111111111111    trivial
ρ21111111111111-1-111111-1-1-1-1    linear of order 2
ρ31-111111111111i-i-1-1-1-1-1i-ii-i    linear of order 4
ρ41-111111111111-ii-1-1-1-1-1-ii-ii    linear of order 4
ρ52222-1-1-1-1-12-1-120022-1-1-10000    orthogonal lifted from S3
ρ62-222-1-1-1-1-12-1-1200-2-21110000    symplectic lifted from Dic3, Schur index 2
ρ73-1-3+3√-3/2-3-3√-3/2-3-3√-3/2-3+3√-3/2300000011ζ6ζ65ζ65-1ζ6ζ3ζ32ζ32ζ3    complex lifted from He3⋊C4
ρ83-1-3-3√-3/2-3+3√-3/2-3+3√-3/2-3-3√-3/2300000011ζ65ζ6ζ6-1ζ65ζ32ζ3ζ3ζ32    complex lifted from He3⋊C4
ρ93-1-3-3√-3/2-3+3√-3/2-3+3√-3/2-3-3√-3/23000000-1-1ζ65ζ6ζ6-1ζ65ζ6ζ65ζ65ζ6    complex lifted from He3⋊C4
ρ103-1-3+3√-3/2-3-3√-3/2-3-3√-3/2-3+3√-3/23000000-1-1ζ6ζ65ζ65-1ζ6ζ65ζ6ζ6ζ65    complex lifted from He3⋊C4
ρ1131-3+3√-3/2-3-3√-3/2-3-3√-3/2-3+3√-3/23000000-iiζ32ζ3ζ31ζ32ζ43ζ3ζ4ζ32ζ43ζ32ζ4ζ3    complex lifted from He3⋊C4
ρ1231-3-3√-3/2-3+3√-3/2-3+3√-3/2-3-3√-3/23000000i-iζ3ζ32ζ321ζ3ζ4ζ32ζ43ζ3ζ4ζ3ζ43ζ32    complex lifted from He3⋊C4
ρ1331-3-3√-3/2-3+3√-3/2-3+3√-3/2-3-3√-3/23000000-iiζ3ζ32ζ321ζ3ζ43ζ32ζ4ζ3ζ43ζ3ζ4ζ32    complex lifted from He3⋊C4
ρ1431-3+3√-3/2-3-3√-3/2-3-3√-3/2-3+3√-3/23000000i-iζ32ζ3ζ31ζ32ζ4ζ3ζ43ζ32ζ4ζ32ζ43ζ3    complex lifted from He3⋊C4
ρ154044444-2-2111-200000000000    orthogonal lifted from C32⋊C4
ρ16404444411-2-2-2100000000000    orthogonal lifted from C32⋊C4
ρ174044-2-2-2-1-3√-3/2-1+3√-3/2-211100000000000    complex lifted from C33⋊C4
ρ184044-2-2-2111-1+3√-3/2-1-3√-3/2-200000000000    complex lifted from C33⋊C4
ρ194044-2-2-2111-1-3√-3/2-1+3√-3/2-200000000000    complex lifted from C33⋊C4
ρ204044-2-2-2-1+3√-3/2-1-3√-3/2-211100000000000    complex lifted from C33⋊C4
ρ2162-3+3√-3-3-3√-33+3√-3/23-3√-3/2-300000000-1-√-3-1+√-3ζ65-1ζ60000    complex faithful
ρ2262-3-3√-3-3+3√-33-3√-3/23+3√-3/2-300000000-1+√-3-1-√-3ζ6-1ζ650000    complex faithful
ρ236-2-3+3√-3-3-3√-33+3√-3/23-3√-3/2-3000000001+√-31-√-3ζ31ζ320000    complex faithful
ρ246-2-3-3√-3-3+3√-33-3√-3/23+3√-3/2-3000000001-√-31+√-3ζ321ζ30000    complex faithful

Permutation representations of He3⋊4Dic3
►On 18 points - transitive group 18T131
Generators in S18
(1 14 13)(2 18 17)(3 16 15)(4 5 6)(7 9 11)

(1 2 3)(4 6 5)(7 9 11)(8 10 12)(13 17 15)(14 18 16)

(1 15 14)(2 13 18)(3 17 16)(4 11 12)(5 9 10)(6 7 8)

(1 2 3)(4 5 6)(7 8 9 10 11 12)(13 14 15 16 17 18)

(1 6)(2 5)(3 4)(7 13 10 16)(8 18 11 15)(9 17 12 14)
 
G:=sub<Sym(18)| (1,14,13)(2,18,17)(3,16,15)(4,5,6)(7,9,11), (1,2,3)(4,6,5)(7,9,11)(8,10,12)(13,17,15)(14,18,16), (1,15,14)(2,13,18)(3,17,16)(4,11,12)(5,9,10)(6,7,8), (1,2,3)(4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18), (1,6)(2,5)(3,4)(7,13,10,16)(8,18,11,15)(9,17,12,14)>;
 
G:=Group( (1,14,13)(2,18,17)(3,16,15)(4,5,6)(7,9,11), (1,2,3)(4,6,5)(7,9,11)(8,10,12)(13,17,15)(14,18,16), (1,15,14)(2,13,18)(3,17,16)(4,11,12)(5,9,10)(6,7,8), (1,2,3)(4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18), (1,6)(2,5)(3,4)(7,13,10,16)(8,18,11,15)(9,17,12,14) );
 
G=PermutationGroup([[(1,14,13),(2,18,17),(3,16,15),(4,5,6),(7,9,11)], [(1,2,3),(4,6,5),(7,9,11),(8,10,12),(13,17,15),(14,18,16)], [(1,15,14),(2,13,18),(3,17,16),(4,11,12),(5,9,10),(6,7,8)], [(1,2,3),(4,5,6),(7,8,9,10,11,12),(13,14,15,16,17,18)], [(1,6),(2,5),(3,4),(7,13,10,16),(8,18,11,15),(9,17,12,14)]])
 
G:=TransitiveGroup(18,131);
 

Matrix representation of He3⋊4Dic3 ►in GL5(𝔽13)

10000
01000
00030
00447
001299
,
10000
01000
00300
00030
00003
,
10000
01000
0012128
00300
001011
,
012000
112000
00100
0012128
00001
,
01000
10000
001105
00555
00202

G:=sub<GL(5,GF(13))| [1,0,0,0,0,0,1,0,0,0,0,0,0,4,12,0,0,3,4,9,0,0,0,7,9],[1,0,0,0,0,0,1,0,0,0,0,0,3,0,0,0,0,0,3,0,0,0,0,0,3],[1,0,0,0,0,0,1,0,0,0,0,0,12,3,10,0,0,12,0,1,0,0,8,0,1],[0,1,0,0,0,12,12,0,0,0,0,0,1,12,0,0,0,0,12,0,0,0,0,8,1],[0,1,0,0,0,1,0,0,0,0,0,0,11,5,2,0,0,0,5,0,0,0,5,5,2] >;
 

He3⋊4Dic3 in GAP, Magma, Sage, TeX 

{\rm He}_3\rtimes_4{\rm Dic}_3
 
% in TeX
 
G:=Group("He3:4Dic3");
 
// GroupNames label
 
G:=SmallGroup(324,113);
 
// by ID
 
G=gap.SmallGroup(324,113);
 
# by ID
 
G:=PCGroup([6,-2,-2,-3,3,-3,-3,12,362,80,2979,1593,1383,2164]);
 
// 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,c*a*c^-1=a*b^-1,d*a*d^-1=a^-1*b,e*a*e^-1=a^-1*b*c^-1,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d^-1=b*c^-1,e*c*e^-1=a^-1*b*c,e*d*e^-1=d^-1>;
 
// generators/relations
 

Export

Subgroup lattice of He3⋊4Dic3 in TeX
Character table of He3⋊4Dic3 in TeX

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