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## G = Dic3⋊D6order 144 = 24·32

### 2nd semidirect product of Dic3 and D6 acting via D6/S3=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — Dic3⋊D6
 Chief series C1 — C3 — C32 — C3×C6 — S3×C6 — C2×S32 — Dic3⋊D6
 Lower central C32 — C3×C6 — Dic3⋊D6
 Upper central C1 — C2 — C22

Generators and relations for Dic3⋊D6
G = < a,b,c,d | a6=c6=d2=1, b2=a3, bab-1=dad=a-1, ac=ca, cbc-1=a3b, bd=db, dcd=c-1 >

Subgroups: 496 in 124 conjugacy classes, 34 normal (12 characteristic)
C1, C2, C2 [×6], C3 [×2], C3, C4 [×2], C22, C22 [×8], S3 [×12], C6 [×2], C6 [×7], C2×C4, D4 [×4], C23 [×2], C32, Dic3 [×2], C12 [×2], D6 [×2], D6 [×18], C2×C6 [×2], C2×C6 [×3], C2×D4, C3×S3 [×2], C3⋊S3 [×2], C3⋊S3, C3×C6, C3×C6, C4×S3 [×2], D12 [×2], C3⋊D4 [×2], C3⋊D4 [×2], C3×D4 [×2], C22×S3 [×5], C3×Dic3 [×2], S32 [×2], S3×C6 [×2], C2×C3⋊S3 [×2], C2×C3⋊S3 [×2], C62, S3×D4 [×2], C6.D6, C3⋊D12 [×2], C3×C3⋊D4 [×2], C2×S32, C22×C3⋊S3, Dic3⋊D6
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×2], C23, D6 [×6], C2×D4, C22×S3 [×2], S32, S3×D4 [×2], C2×S32, Dic3⋊D6

Character table of Dic3⋊D6

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 4A 4B 6A 6B 6C 6D 6E 6F 6G 6H 6I 12A 12B size 1 1 2 6 6 9 9 18 2 2 4 6 6 2 2 4 4 4 4 4 12 12 12 12 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 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 1 1 1 1 -1 -1 1 linear of order 2 ρ5 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 ρ6 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 ρ7 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 ρ8 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 ρ9 2 2 2 -2 0 0 0 0 2 -1 -1 -2 0 -1 2 -1 -1 -1 2 -1 1 0 0 1 orthogonal lifted from D6 ρ10 2 2 -2 0 -2 0 0 0 -1 2 -1 0 2 2 -1 -1 1 -2 1 1 0 1 -1 0 orthogonal lifted from D6 ρ11 2 2 -2 -2 0 0 0 0 2 -1 -1 2 0 -1 2 -1 1 1 -2 1 1 0 0 -1 orthogonal lifted from D6 ρ12 2 2 -2 0 2 0 0 0 -1 2 -1 0 -2 2 -1 -1 1 -2 1 1 0 -1 1 0 orthogonal lifted from D6 ρ13 2 -2 0 0 0 -2 2 0 2 2 2 0 0 -2 -2 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ14 2 -2 0 0 0 2 -2 0 2 2 2 0 0 -2 -2 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ15 2 2 2 0 -2 0 0 0 -1 2 -1 0 -2 2 -1 -1 -1 2 -1 -1 0 1 1 0 orthogonal lifted from D6 ρ16 2 2 -2 2 0 0 0 0 2 -1 -1 -2 0 -1 2 -1 1 1 -2 1 -1 0 0 1 orthogonal lifted from D6 ρ17 2 2 2 0 2 0 0 0 -1 2 -1 0 2 2 -1 -1 -1 2 -1 -1 0 -1 -1 0 orthogonal lifted from S3 ρ18 2 2 2 2 0 0 0 0 2 -1 -1 2 0 -1 2 -1 -1 -1 2 -1 -1 0 0 -1 orthogonal lifted from S3 ρ19 4 -4 0 0 0 0 0 0 -2 -2 1 0 0 2 2 -1 -3 0 0 3 0 0 0 0 orthogonal faithful ρ20 4 -4 0 0 0 0 0 0 -2 4 -2 0 0 -4 2 2 0 0 0 0 0 0 0 0 orthogonal lifted from S3×D4 ρ21 4 4 -4 0 0 0 0 0 -2 -2 1 0 0 -2 -2 1 -1 2 2 -1 0 0 0 0 orthogonal lifted from C2×S32 ρ22 4 -4 0 0 0 0 0 0 -2 -2 1 0 0 2 2 -1 3 0 0 -3 0 0 0 0 orthogonal faithful ρ23 4 -4 0 0 0 0 0 0 4 -2 -2 0 0 2 -4 2 0 0 0 0 0 0 0 0 orthogonal lifted from S3×D4 ρ24 4 4 4 0 0 0 0 0 -2 -2 1 0 0 -2 -2 1 1 -2 -2 1 0 0 0 0 orthogonal lifted from S32

Permutation representations of Dic3⋊D6
On 12 points - transitive group 12T81
Generators in S12
(1 2 3 4 5 6)(7 8 9 10 11 12)
(1 8 4 11)(2 7 5 10)(3 12 6 9)
(1 5 3)(2 6 4)(7 12 11 10 9 8)
(1 3)(4 6)(8 12)(9 11)

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

G:=Group( (1,2,3,4,5,6)(7,8,9,10,11,12), (1,8,4,11)(2,7,5,10)(3,12,6,9), (1,5,3)(2,6,4)(7,12,11,10,9,8), (1,3)(4,6)(8,12)(9,11) );

G=PermutationGroup([(1,2,3,4,5,6),(7,8,9,10,11,12)], [(1,8,4,11),(2,7,5,10),(3,12,6,9)], [(1,5,3),(2,6,4),(7,12,11,10,9,8)], [(1,3),(4,6),(8,12),(9,11)])

G:=TransitiveGroup(12,81);

On 24 points - transitive group 24T269
Generators in S24
(1 2 3 4 5 6)(7 8 9 10 11 12)(13 14 15 16 17 18)(19 20 21 22 23 24)
(1 20 4 23)(2 19 5 22)(3 24 6 21)(7 14 10 17)(8 13 11 16)(9 18 12 15)
(1 11 3 7 5 9)(2 12 4 8 6 10)(13 24 17 22 15 20)(14 19 18 23 16 21)
(1 6)(2 5)(3 4)(7 12)(8 11)(9 10)(13 16)(14 15)(17 18)(19 22)(20 21)(23 24)

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

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

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

G:=TransitiveGroup(24,269);

On 24 points - transitive group 24T270
Generators in S24
(1 2 3 4 5 6)(7 8 9 10 11 12)(13 14 15 16 17 18)(19 20 21 22 23 24)
(1 8 4 11)(2 7 5 10)(3 12 6 9)(13 23 16 20)(14 22 17 19)(15 21 18 24)
(1 5 3)(2 6 4)(7 12 11 10 9 8)(13 18 17 16 15 14)(19 23 21)(20 24 22)
(1 19)(2 24)(3 23)(4 22)(5 21)(6 20)(7 15)(8 14)(9 13)(10 18)(11 17)(12 16)

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

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

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

G:=TransitiveGroup(24,270);

On 24 points - transitive group 24T271
Generators in S24
(1 2 3 4 5 6)(7 8 9 10 11 12)(13 14 15 16 17 18)(19 20 21 22 23 24)
(1 20 4 23)(2 19 5 22)(3 24 6 21)(7 14 10 17)(8 13 11 16)(9 18 12 15)
(1 11 3 7 5 9)(2 12 4 8 6 10)(13 24 17 22 15 20)(14 19 18 23 16 21)
(1 9)(2 8)(3 7)(4 12)(5 11)(6 10)(13 19)(14 24)(15 23)(16 22)(17 21)(18 20)

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

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

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

G:=TransitiveGroup(24,271);

Polynomial with Galois group Dic3⋊D6 over ℚ
actionf(x)Disc(f)
12T81x12-24x10+216x8-902x6+1752x4-1368x2+200239·312·56·76·236

Matrix representation of Dic3⋊D6 in GL4(ℤ) generated by

 1 1 0 0 -1 0 0 0 0 0 0 -1 0 0 1 1
,
 0 0 1 0 0 0 0 1 -1 0 0 0 0 -1 0 0
,
 -1 -1 0 0 1 0 0 0 0 0 1 1 0 0 -1 0
,
 1 1 0 0 0 -1 0 0 0 0 1 1 0 0 0 -1
G:=sub<GL(4,Integers())| [1,-1,0,0,1,0,0,0,0,0,0,1,0,0,-1,1],[0,0,-1,0,0,0,0,-1,1,0,0,0,0,1,0,0],[-1,1,0,0,-1,0,0,0,0,0,1,-1,0,0,1,0],[1,0,0,0,1,-1,0,0,0,0,1,0,0,0,1,-1] >;

Dic3⋊D6 in GAP, Magma, Sage, TeX

{\rm Dic}_3\rtimes D_6
% in TeX

G:=Group("Dic3:D6");
// GroupNames label

G:=SmallGroup(144,154);
// by ID

G=gap.SmallGroup(144,154);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-3,218,116,490,3461]);
// Polycyclic

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

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