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## G = He3⋊D6order 324 = 22·34

### The semidirect product of He3 and D6 acting faithfully

Aliases: He3⋊D6, C331D6, C3≀S3⋊C2, C3≀C3⋊C22, C32.1S32, C33⋊C6⋊C2, C33⋊S3⋊C2, He3⋊C21S3, C33⋊C21S3, 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 [×3], C3, C3 [×5], C22, S3 [×12], C6 [×6], C9, C32, C32 [×5], D6 [×6], D9, C3×S3 [×10], C3⋊S3 [×7], C3×C6, He3, 3- 1+2, C33, S32 [×5], C2×C3⋊S3, C32⋊C6 [×2], 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 [×3], C22, S3 [×2], D6 [×2], 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
(2 6 7)(4 12 10)(13 15 17)(14 16 18)
(1 5 8)(2 6 7)(3 11 9)(4 12 10)(13 17 15)(14 18 16)
(1 18 12)(2 11 15)(3 17 7)(4 8 14)(5 16 10)(6 9 13)
(1 2)(3 4)(5 6)(7 8)(9 10)(11 12)(13 14 15 16 17 18)
(1 2)(3 10)(4 9)(5 7)(6 8)(11 12)(13 16)(14 15)(17 18)

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

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

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

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

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

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

G:=TransitiveGroup(18,137);

On 27 points - transitive group 27T121
Generators in S27
(1 25 12)(2 23 10)(3 27 14)(4 15 21)(5 13 19)(6 11 17)(7 20 24)(8 18 22)(9 16 26)
(1 8 4)(2 9 5)(3 7 6)(10 26 19)(11 27 20)(12 22 21)(13 23 16)(14 24 17)(15 25 18)
(1 25 22)(2 10 13)(4 15 12)(5 19 16)(8 18 21)(9 26 23)(11 27 20)(14 17 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)
(2 3)(4 8)(5 7)(6 9)(10 24)(11 23)(12 22)(13 27)(14 26)(15 25)(16 20)(17 19)

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

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

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

G:=TransitiveGroup(27,121);

On 27 points - transitive group 27T128
Generators in S27
(1 7 5)(2 4 8)(3 9 6)(10 23 17)(11 13 15)(12 25 19)(14 27 21)(16 18 20)(22 24 26)
(1 3 2)(4 7 9)(5 6 8)(10 25 21)(11 26 16)(12 27 17)(13 22 18)(14 23 19)(15 24 20)
(1 26 23)(2 11 14)(3 16 19)(4 22 12)(5 15 25)(6 24 21)(7 18 27)(8 20 10)(9 13 17)
(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 7)(5 6)(10 16)(11 21)(12 20)(13 19)(14 18)(15 17)(22 23)(24 27)(25 26)

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

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

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

G:=TransitiveGroup(27,128);

On 27 points - transitive group 27T129
Generators in S27
(1 8 5)(2 9 6)(3 7 4)(10 27 17)(12 23 19)(14 25 21)
(1 5 8)(2 6 9)(3 4 7)(10 27 17)(11 22 18)(12 23 19)(13 24 20)(14 25 21)(15 26 16)
(1 12 15)(2 27 24)(3 21 18)(4 14 11)(5 23 26)(6 17 20)(7 25 22)(8 19 16)(9 10 13)
(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)
(2 3)(4 9)(5 8)(6 7)(10 22)(11 27)(12 26)(13 25)(14 24)(15 23)(16 19)(17 18)(20 21)

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

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

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

G:=TransitiveGroup(27,129);

Polynomial with Galois group He3⋊D6 over ℚ
actionf(x)Disc(f)
9T24x9-2x8-17x7+33x6+90x5-183x4-161x3+379x2+36x-19354·1392·1973

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

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