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G = He34Dic3order 324 = 22·34

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

non-abelian, soluble

Aliases: He34Dic3, 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

C1C3C3×He3 — He34Dic3
C1C3C32C3×He3C3×He3⋊C2 — He34Dic3
C3×He3 — He34Dic3
C1C3

Generators and relations for He34Dic3
 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 >

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

Character table of He34Dic3

 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 He34Dic3
On 18 points - transitive group 18T131
Generators in S18
(1 2 3)(4 18 17)(5 16 15)(6 14 13)(8 10 12)
(1 3 2)(4 5 6)(7 9 11)(8 10 12)(13 17 15)(14 18 16)
(1 10 11)(2 8 9)(3 12 7)(4 13 18)(5 17 16)(6 15 14)
(1 2 3)(4 5 6)(7 8 9 10 11 12)(13 14 15 16 17 18)
(1 5)(2 4)(3 6)(7 18 10 15)(8 17 11 14)(9 16 12 13)

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

G:=Group( (1,2,3)(4,18,17)(5,16,15)(6,14,13)(8,10,12), (1,3,2)(4,5,6)(7,9,11)(8,10,12)(13,17,15)(14,18,16), (1,10,11)(2,8,9)(3,12,7)(4,13,18)(5,17,16)(6,15,14), (1,2,3)(4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18), (1,5)(2,4)(3,6)(7,18,10,15)(8,17,11,14)(9,16,12,13) );

G=PermutationGroup([(1,2,3),(4,18,17),(5,16,15),(6,14,13),(8,10,12)], [(1,3,2),(4,5,6),(7,9,11),(8,10,12),(13,17,15),(14,18,16)], [(1,10,11),(2,8,9),(3,12,7),(4,13,18),(5,17,16),(6,15,14)], [(1,2,3),(4,5,6),(7,8,9,10,11,12),(13,14,15,16,17,18)], [(1,5),(2,4),(3,6),(7,18,10,15),(8,17,11,14),(9,16,12,13)])

G:=TransitiveGroup(18,131);

Matrix representation of He34Dic3 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] >;

He34Dic3 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 He34Dic3 in TeX
Character table of He34Dic3 in TeX

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