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## G = He3⋊4Dic3order 324 = 22·34

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

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

 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 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 >

Character table of He34Dic3

 class 1 2 3A 3B 3C 3D 3E 3F 3G 3H 3I 3J 3K 4A 4B 6A 6B 6C 6D 6E 12A 12B 12C 12D size 1 9 1 1 2 2 2 12 12 12 12 12 12 27 27 9 9 18 18 18 27 27 27 27 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 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 1 -1 -1 1 1 1 1 1 -1 -1 -1 -1 linear of order 2 ρ3 1 -1 1 1 1 1 1 1 1 1 1 1 1 i -i -1 -1 -1 -1 -1 i -i i -i linear of order 4 ρ4 1 -1 1 1 1 1 1 1 1 1 1 1 1 -i i -1 -1 -1 -1 -1 -i i -i i linear of order 4 ρ5 2 2 2 2 -1 -1 -1 -1 -1 2 -1 -1 2 0 0 2 2 -1 -1 -1 0 0 0 0 orthogonal lifted from S3 ρ6 2 -2 2 2 -1 -1 -1 -1 -1 2 -1 -1 2 0 0 -2 -2 1 1 1 0 0 0 0 symplectic lifted from Dic3, Schur index 2 ρ7 3 -1 -3+3√-3/2 -3-3√-3/2 -3-3√-3/2 -3+3√-3/2 3 0 0 0 0 0 0 1 1 ζ6 ζ65 ζ65 -1 ζ6 ζ3 ζ32 ζ32 ζ3 complex lifted from He3⋊C4 ρ8 3 -1 -3-3√-3/2 -3+3√-3/2 -3+3√-3/2 -3-3√-3/2 3 0 0 0 0 0 0 1 1 ζ65 ζ6 ζ6 -1 ζ65 ζ32 ζ3 ζ3 ζ32 complex lifted from He3⋊C4 ρ9 3 -1 -3-3√-3/2 -3+3√-3/2 -3+3√-3/2 -3-3√-3/2 3 0 0 0 0 0 0 -1 -1 ζ65 ζ6 ζ6 -1 ζ65 ζ6 ζ65 ζ65 ζ6 complex lifted from He3⋊C4 ρ10 3 -1 -3+3√-3/2 -3-3√-3/2 -3-3√-3/2 -3+3√-3/2 3 0 0 0 0 0 0 -1 -1 ζ6 ζ65 ζ65 -1 ζ6 ζ65 ζ6 ζ6 ζ65 complex lifted from He3⋊C4 ρ11 3 1 -3+3√-3/2 -3-3√-3/2 -3-3√-3/2 -3+3√-3/2 3 0 0 0 0 0 0 -i i ζ32 ζ3 ζ3 1 ζ32 ζ43ζ3 ζ4ζ32 ζ43ζ32 ζ4ζ3 complex lifted from He3⋊C4 ρ12 3 1 -3-3√-3/2 -3+3√-3/2 -3+3√-3/2 -3-3√-3/2 3 0 0 0 0 0 0 i -i ζ3 ζ32 ζ32 1 ζ3 ζ4ζ32 ζ43ζ3 ζ4ζ3 ζ43ζ32 complex lifted from He3⋊C4 ρ13 3 1 -3-3√-3/2 -3+3√-3/2 -3+3√-3/2 -3-3√-3/2 3 0 0 0 0 0 0 -i i ζ3 ζ32 ζ32 1 ζ3 ζ43ζ32 ζ4ζ3 ζ43ζ3 ζ4ζ32 complex lifted from He3⋊C4 ρ14 3 1 -3+3√-3/2 -3-3√-3/2 -3-3√-3/2 -3+3√-3/2 3 0 0 0 0 0 0 i -i ζ32 ζ3 ζ3 1 ζ32 ζ4ζ3 ζ43ζ32 ζ4ζ32 ζ43ζ3 complex lifted from He3⋊C4 ρ15 4 0 4 4 4 4 4 -2 -2 1 1 1 -2 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C32⋊C4 ρ16 4 0 4 4 4 4 4 1 1 -2 -2 -2 1 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C32⋊C4 ρ17 4 0 4 4 -2 -2 -2 -1-3√-3/2 -1+3√-3/2 -2 1 1 1 0 0 0 0 0 0 0 0 0 0 0 complex lifted from C33⋊C4 ρ18 4 0 4 4 -2 -2 -2 1 1 1 -1+3√-3/2 -1-3√-3/2 -2 0 0 0 0 0 0 0 0 0 0 0 complex lifted from C33⋊C4 ρ19 4 0 4 4 -2 -2 -2 1 1 1 -1-3√-3/2 -1+3√-3/2 -2 0 0 0 0 0 0 0 0 0 0 0 complex lifted from C33⋊C4 ρ20 4 0 4 4 -2 -2 -2 -1+3√-3/2 -1-3√-3/2 -2 1 1 1 0 0 0 0 0 0 0 0 0 0 0 complex lifted from C33⋊C4 ρ21 6 2 -3+3√-3 -3-3√-3 3+3√-3/2 3-3√-3/2 -3 0 0 0 0 0 0 0 0 -1-√-3 -1+√-3 ζ65 -1 ζ6 0 0 0 0 complex faithful ρ22 6 2 -3-3√-3 -3+3√-3 3-3√-3/2 3+3√-3/2 -3 0 0 0 0 0 0 0 0 -1+√-3 -1-√-3 ζ6 -1 ζ65 0 0 0 0 complex faithful ρ23 6 -2 -3+3√-3 -3-3√-3 3+3√-3/2 3-3√-3/2 -3 0 0 0 0 0 0 0 0 1+√-3 1-√-3 ζ3 1 ζ32 0 0 0 0 complex faithful ρ24 6 -2 -3-3√-3 -3+3√-3 3-3√-3/2 3+3√-3/2 -3 0 0 0 0 0 0 0 0 1-√-3 1+√-3 ζ32 1 ζ3 0 0 0 0 complex faithful

Permutation representations of He34Dic3
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 He34Dic3 in GL5(𝔽13)

 1 0 0 0 0 0 1 0 0 0 0 0 0 3 0 0 0 4 4 7 0 0 12 9 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 12 8 0 0 3 0 0 0 0 10 1 1
,
 0 12 0 0 0 1 12 0 0 0 0 0 1 0 0 0 0 12 12 8 0 0 0 0 1
,
 0 1 0 0 0 1 0 0 0 0 0 0 11 0 5 0 0 5 5 5 0 0 2 0 2

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

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