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G = He3:4Dic3order 324 = 22·34

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

non-abelian, soluble

Aliases: He3:4Dic3, C3:(He3:C4), (C3xHe3):2C4, He3:C2.2S3, C3.2(C33:C4), C32.2(C32:C4), (C3xHe3:C2).2C2, SmallGroup(324,113)

Series: Derived Chief Lower central Upper central

C1C3C3xHe3 — He3:4Dic3
C1C3C32C3xHe3C3xHe3:C2 — He3:4Dic3
C3xHe3 — He3:4Dic3
C1C3

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 >

Subgroups: 308 in 50 conjugacy classes, 9 normal (all characteristic)
Quotients: C1, C2, C4, S3, Dic3, C32:C4, He3:C4, C33:C4, He3:4Dic3
9C2
2C3
6C3
6C3
12C3
12C3
27C4
6S3
6S3
9C6
9C6
18C6
2C32
2C32
4C32
4C32
6C32
6C32
6C32
6C32
6C32
6C32
9Dic3
27C12
6C3xS3
6C3xS3
6C3xS3
6C3xS3
6C3xS3
6C3xS3
6C3xS3
6C3xS3
9C3xC6
2C33
2C33
4He3
4He3
9C3xDic3
6S3xC32
6S3xC32
3He3:C4

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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