He3:D6 - GroupNames

G = He₃⋊D₆ order 324 = 2²·3⁴

The semidirect product of He₃ and D₆ acting faithfully

non-abelian, supersoluble, monomial, rational

Aliases: He₃⋊D₆, C₃³⋊₁D₆, C₃≀S₃⋊C₂, C₃≀C₃⋊C₂², C₃².₁S₃², C₃³⋊C₆⋊C₂, C₃³⋊S₃⋊C₂, He₃⋊C₂⋊₁S₃, C₃³⋊C₂⋊₁S₃, C₃.₂(C₃²⋊D₆), SmallGroup(324,39)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃² — C₃≀C₃ — He₃⋊D₆

Chief series

C₁ — C₃ — C₃² — He₃ — C₃≀C₃ — C₃≀S₃ — He₃⋊D₆

Lower central

C₃≀C₃ — He₃⋊D₆

Upper central

C₁

Generators and relations for He₃⋊D₆
G = < a,b,c,d,e | a³=b³=c³=d⁶=e²=1, ab=ba, cac^-1=eae=ab^-1, dad^-1=a^-1b, bc=cb, bd=db, ebe=b^-1, dcd^-1=ece=a^-1c^-1, ede=d^-1 >

Subgroups: 848 in 84 conjugacy classes, 12 normal (all characteristic)
C₁, C₂, C₃, C₃, C₂², S₃, C₆, C₉, C₃², C₃², D₆, D₉, C₃×S₃, C₃⋊S₃, C₃×C₆, He₃, 3_-¹⁺², C₃³, S₃², C₂×C₃⋊S₃, C₃²⋊C₆, C₉⋊C₆, He₃⋊C₂, S₃×C₃², C₃×C₃⋊S₃, C₃³⋊C₂, C₃≀C₃, C₃²⋊D₆, S₃×C₃⋊S₃, C₃≀S₃, C₃³⋊C₆, C₃³⋊S₃, He₃⋊D₆
Quotients: C₁, C₂, C₂², S₃, D₆, S₃², C₃²⋊D₆, He₃⋊D₆

Character table of He₃⋊D₆

class	1	2A	2B	2C	3A	3B	3C	3D	3E	3F	6A	6B	6C	6D	6E	6F	9
size	1	9	27	27	2	6	6	6	6	18	18	18	18	18	54	54	36
ρ₁	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	linear of order 2
ρ₃	1	1	-1	-1	1	1	1	1	1	1	1	1	1	1	-1	-1	1	linear of order 2
ρ₄	1	-1	-1	1	1	1	1	1	1	1	-1	-1	-1	-1	-1	1	1	linear of order 2
ρ₅	2	2	0	0	2	-1	-1	-1	2	2	2	-1	-1	-1	0	0	-1	orthogonal lifted from S₃
ρ₆	2	0	2	0	2	2	2	2	2	-1	0	0	0	0	-1	0	-1	orthogonal lifted from S₃
ρ₇	2	0	-2	0	2	2	2	2	2	-1	0	0	0	0	1	0	-1	orthogonal lifted from D₆
ρ₈	2	-2	0	0	2	-1	-1	-1	2	2	-2	1	1	1	0	0	-1	orthogonal lifted from D₆
ρ₉	4	0	0	0	4	-2	-2	-2	4	-2	0	0	0	0	0	0	1	orthogonal lifted from S₃²
ρ₁₀	6	2	0	0	-3	3	-3	0	0	0	-1	-1	2	-1	0	0	0	orthogonal faithful
ρ₁₁	6	-2	0	0	-3	-3	0	3	0	0	1	1	1	-2	0	0	0	orthogonal faithful
ρ₁₂	6	-2	0	0	-3	0	3	-3	0	0	1	-2	1	1	0	0	0	orthogonal faithful
ρ₁₃	6	0	0	2	6	0	0	0	-3	0	0	0	0	0	0	-1	0	orthogonal lifted from C₃²⋊D₆
ρ₁₄	6	2	0	0	-3	-3	0	3	0	0	-1	-1	-1	2	0	0	0	orthogonal faithful
ρ₁₅	6	0	0	-2	6	0	0	0	-3	0	0	0	0	0	0	1	0	orthogonal lifted from C₃²⋊D₆
ρ₁₆	6	-2	0	0	-3	3	-3	0	0	0	1	1	-2	1	0	0	0	orthogonal faithful
ρ₁₇	6	2	0	0	-3	0	3	-3	0	0	-1	2	-1	-1	0	0	0	orthogonal faithful

Permutation representations of He₃⋊D₆
►On 9 points - transitive group 9T24

Generators in S₉

(1 2 3)(5 9 7)

(1 3 2)(4 8 6)(5 9 7)

(1 7 4)(2 9 6)(3 5 8)

(1 2 3)(4 5 6 7 8 9)

(1 3)(4 9)(5 8)(6 7)

G:=sub<Sym(9)| (1,2,3)(5,9,7), (1,3,2)(4,8,6)(5,9,7), (1,7,4)(2,9,6)(3,5,8), (1,2,3)(4,5,6,7,8,9), (1,3)(4,9)(5,8)(6,7)>;

G:=Group( (1,2,3)(5,9,7), (1,3,2)(4,8,6)(5,9,7), (1,7,4)(2,9,6)(3,5,8), (1,2,3)(4,5,6,7,8,9), (1,3)(4,9)(5,8)(6,7) );

G=PermutationGroup([[(1,2,3),(5,9,7)], [(1,3,2),(4,8,6),(5,9,7)], [(1,7,4),(2,9,6),(3,5,8)], [(1,2,3),(4,5,6,7,8,9)], [(1,3),(4,9),(5,8),(6,7)]])

G:=TransitiveGroup(9,24);

►On 18 points - transitive group 18T129

Generators in S₁₈

(1 10 12)(4 6 8)(13 15 17)(14 16 18)

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

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

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

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

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

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

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

G:=TransitiveGroup(18,129);

►On 18 points - transitive group 18T136

Generators in S₁₈

(1 5 4)(2 6 3)(7 9 11)(14 16 18)

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

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

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

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

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

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

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

G:=TransitiveGroup(18,136);

►On 18 points - transitive group 18T137

Generators in S₁₈

(1 2 3)(4 5 6)(7 11 9)(14 18 16)

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

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

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

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

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

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

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

G:=TransitiveGroup(18,137);

►On 27 points - transitive group 27T121

Generators in S₂₇

(1 11 17)(2 15 21)(3 13 19)(4 20 26)(5 18 24)(6 16 22)(7 23 14)(8 27 12)(9 25 10)

(1 7 4)(2 8 5)(3 9 6)(10 22 19)(11 23 20)(12 24 21)(13 25 16)(14 26 17)(15 27 18)

(2 15 12)(3 19 16)(5 18 21)(6 22 25)(8 27 24)(9 10 13)(11 23 20)(14 17 26)

(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)

(1 9)(2 8)(3 7)(4 6)(10 14)(11 13)(16 23)(17 22)(18 27)(19 26)(20 25)(21 24)

G:=sub<Sym(27)| (1,11,17)(2,15,21)(3,13,19)(4,20,26)(5,18,24)(6,16,22)(7,23,14)(8,27,12)(9,25,10), (1,7,4)(2,8,5)(3,9,6)(10,22,19)(11,23,20)(12,24,21)(13,25,16)(14,26,17)(15,27,18), (2,15,12)(3,19,16)(5,18,21)(6,22,25)(8,27,24)(9,10,13)(11,23,20)(14,17,26), (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), (1,9)(2,8)(3,7)(4,6)(10,14)(11,13)(16,23)(17,22)(18,27)(19,26)(20,25)(21,24)>;

G:=Group( (1,11,17)(2,15,21)(3,13,19)(4,20,26)(5,18,24)(6,16,22)(7,23,14)(8,27,12)(9,25,10), (1,7,4)(2,8,5)(3,9,6)(10,22,19)(11,23,20)(12,24,21)(13,25,16)(14,26,17)(15,27,18), (2,15,12)(3,19,16)(5,18,21)(6,22,25)(8,27,24)(9,10,13)(11,23,20)(14,17,26), (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), (1,9)(2,8)(3,7)(4,6)(10,14)(11,13)(16,23)(17,22)(18,27)(19,26)(20,25)(21,24) );

G=PermutationGroup([[(1,11,17),(2,15,21),(3,13,19),(4,20,26),(5,18,24),(6,16,22),(7,23,14),(8,27,12),(9,25,10)], [(1,7,4),(2,8,5),(3,9,6),(10,22,19),(11,23,20),(12,24,21),(13,25,16),(14,26,17),(15,27,18)], [(2,15,12),(3,19,16),(5,18,21),(6,22,25),(8,27,24),(9,10,13),(11,23,20),(14,17,26)], [(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)], [(1,9),(2,8),(3,7),(4,6),(10,14),(11,13),(16,23),(17,22),(18,27),(19,26),(20,25),(21,24)]])

G:=TransitiveGroup(27,121);

►On 27 points - transitive group 27T128

Generators in S₂₇

(1 7 4)(2 5 9)(3 8 6)(10 16 26)(11 13 15)(12 18 22)(14 20 24)(17 19 21)(23 25 27)

(1 3 2)(4 6 9)(5 7 8)(10 18 24)(11 19 25)(12 20 26)(13 21 27)(14 16 22)(15 17 23)

(1 15 12)(2 23 26)(3 17 20)(4 27 14)(5 11 24)(6 13 16)(7 19 10)(8 25 18)(9 21 22)

(4 5)(6 7)(8 9)(10 11 12 13 14 15)(16 17 18 19 20 21)(22 23 24 25 26 27)

(2 3)(4 6)(5 7)(10 13)(11 12)(14 15)(16 23)(17 22)(18 27)(19 26)(20 25)(21 24)

G:=sub<Sym(27)| (1,7,4)(2,5,9)(3,8,6)(10,16,26)(11,13,15)(12,18,22)(14,20,24)(17,19,21)(23,25,27), (1,3,2)(4,6,9)(5,7,8)(10,18,24)(11,19,25)(12,20,26)(13,21,27)(14,16,22)(15,17,23), (1,15,12)(2,23,26)(3,17,20)(4,27,14)(5,11,24)(6,13,16)(7,19,10)(8,25,18)(9,21,22), (4,5)(6,7)(8,9)(10,11,12,13,14,15)(16,17,18,19,20,21)(22,23,24,25,26,27), (2,3)(4,6)(5,7)(10,13)(11,12)(14,15)(16,23)(17,22)(18,27)(19,26)(20,25)(21,24)>;

G:=Group( (1,7,4)(2,5,9)(3,8,6)(10,16,26)(11,13,15)(12,18,22)(14,20,24)(17,19,21)(23,25,27), (1,3,2)(4,6,9)(5,7,8)(10,18,24)(11,19,25)(12,20,26)(13,21,27)(14,16,22)(15,17,23), (1,15,12)(2,23,26)(3,17,20)(4,27,14)(5,11,24)(6,13,16)(7,19,10)(8,25,18)(9,21,22), (4,5)(6,7)(8,9)(10,11,12,13,14,15)(16,17,18,19,20,21)(22,23,24,25,26,27), (2,3)(4,6)(5,7)(10,13)(11,12)(14,15)(16,23)(17,22)(18,27)(19,26)(20,25)(21,24) );

G=PermutationGroup([[(1,7,4),(2,5,9),(3,8,6),(10,16,26),(11,13,15),(12,18,22),(14,20,24),(17,19,21),(23,25,27)], [(1,3,2),(4,6,9),(5,7,8),(10,18,24),(11,19,25),(12,20,26),(13,21,27),(14,16,22),(15,17,23)], [(1,15,12),(2,23,26),(3,17,20),(4,27,14),(5,11,24),(6,13,16),(7,19,10),(8,25,18),(9,21,22)], [(4,5),(6,7),(8,9),(10,11,12,13,14,15),(16,17,18,19,20,21),(22,23,24,25,26,27)], [(2,3),(4,6),(5,7),(10,13),(11,12),(14,15),(16,23),(17,22),(18,27),(19,26),(20,25),(21,24)]])

G:=TransitiveGroup(27,128);

►On 27 points - transitive group 27T129

Generators in S₂₇

(1 8 5)(2 9 6)(3 7 4)(11 16 23)(13 18 25)(15 20 27)

(1 5 8)(2 6 9)(3 4 7)(10 21 22)(11 16 23)(12 17 24)(13 18 25)(14 19 26)(15 20 27)

(1 23 26)(2 15 12)(3 18 21)(4 25 22)(5 11 14)(6 20 17)(7 13 10)(8 16 19)(9 27 24)

(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)

(1 3)(4 8)(5 7)(6 9)(10 16)(11 21)(12 20)(13 19)(14 18)(15 17)(22 23)(24 27)(25 26)

G:=sub<Sym(27)| (1,8,5)(2,9,6)(3,7,4)(11,16,23)(13,18,25)(15,20,27), (1,5,8)(2,6,9)(3,4,7)(10,21,22)(11,16,23)(12,17,24)(13,18,25)(14,19,26)(15,20,27), (1,23,26)(2,15,12)(3,18,21)(4,25,22)(5,11,14)(6,20,17)(7,13,10)(8,16,19)(9,27,24), (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), (1,3)(4,8)(5,7)(6,9)(10,16)(11,21)(12,20)(13,19)(14,18)(15,17)(22,23)(24,27)(25,26)>;

G:=Group( (1,8,5)(2,9,6)(3,7,4)(11,16,23)(13,18,25)(15,20,27), (1,5,8)(2,6,9)(3,4,7)(10,21,22)(11,16,23)(12,17,24)(13,18,25)(14,19,26)(15,20,27), (1,23,26)(2,15,12)(3,18,21)(4,25,22)(5,11,14)(6,20,17)(7,13,10)(8,16,19)(9,27,24), (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), (1,3)(4,8)(5,7)(6,9)(10,16)(11,21)(12,20)(13,19)(14,18)(15,17)(22,23)(24,27)(25,26) );

G=PermutationGroup([[(1,8,5),(2,9,6),(3,7,4),(11,16,23),(13,18,25),(15,20,27)], [(1,5,8),(2,6,9),(3,4,7),(10,21,22),(11,16,23),(12,17,24),(13,18,25),(14,19,26),(15,20,27)], [(1,23,26),(2,15,12),(3,18,21),(4,25,22),(5,11,14),(6,20,17),(7,13,10),(8,16,19),(9,27,24)], [(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)], [(1,3),(4,8),(5,7),(6,9),(10,16),(11,21),(12,20),(13,19),(14,18),(15,17),(22,23),(24,27),(25,26)]])

G:=TransitiveGroup(27,129);

Polynomial with Galois group He₃⋊D₆ over ℚ

action	f(x)	Disc(f)
9T24	x⁹-2x⁸-17x⁷+33x⁶+90x⁵-183x⁴-161x³+379x²+36x-193	5⁴·139²·197³

Matrix representation of He₃⋊D₆ ►in GL₆(ℤ)

-1	-1	0	0	0	0
1	0	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	0	1
0	0	0	0	-1	-1

0	1	0	0	0	0
-1	-1	0	0	0	0
0	0	0	1	0	0
0	0	-1	-1	0	0
0	0	0	0	0	1
0	0	0	0	-1	-1

0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	1	0
0	0	0	0	0	1
1	0	0	0	0	0
0	1	0	0	0	0

0	1	0	0	0	0
-1	-1	0	0	0	0
0	0	0	0	1	0
0	0	0	0	0	1
0	0	1	0	0	0
0	0	0	1	0	0

0	1	0	0	0	0
1	0	0	0	0	0
0	0	0	0	1	0
0	0	0	0	-1	-1
0	0	1	0	0	0
0	0	-1	-1	0	0

G:=sub<GL(6,Integers())| [-1,1,0,0,0,0,-1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,0,0,0,0,1,-1],[0,-1,0,0,0,0,1,-1,0,0,0,0,0,0,0,-1,0,0,0,0,1,-1,0,0,0,0,0,0,0,-1,0,0,0,0,1,-1],[0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0],[0,-1,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,-1,0,0,1,-1,0,0,0,0,0,-1,0,0] >;

He₃⋊D₆ in GAP, Magma, Sage, TeX

{\rm He}_3\rtimes D_6

% in TeX

G:=Group("He3:D6");

// GroupNames label

G:=SmallGroup(324,39);

// by ID

G=gap.SmallGroup(324,39);

# by ID

G:=PCGroup([6,-2,-2,-3,-3,-3,-3,80,579,303,5404,1090,382,3899]);

// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^6=e^2=1,a*b=b*a,c*a*c^-1=e*a*e=a*b^-1,d*a*d^-1=a^-1*b,b*c=c*b,b*d=d*b,e*b*e=b^-1,d*c*d^-1=e*c*e=a^-1*c^-1,e*d*e=d^-1>;

// generators/relations

Export

Character table of He₃⋊D₆ in TeX

G = He3⋊D6 order 324 = 22·34

The semidirect product of He3 and D6 acting faithfully

G = He₃⋊D₆ order 324 = 2²·3⁴

The semidirect product of He₃ and D₆ acting faithfully