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G = Dic3⋊D6order 144 = 24·32

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

metabelian, supersoluble, monomial, rational

Aliases: D63D6, Dic32D6, C622C22, C3⋊S33D4, C223S32, C33(S3×D4), (C2×C6)⋊5D6, C3⋊D42S3, C327(C2×D4), C3⋊D126C2, (S3×C6)⋊4C22, C6.D63C2, C6.18(C22×S3), (C3×C6).18C23, (C3×Dic3)⋊2C22, (C2×S32)⋊4C2, C2.18(C2×S32), (C3×C3⋊D4)⋊4C2, (C22×C3⋊S3)⋊2C2, (C2×C3⋊S3)⋊4C22, Hol(Dic3), SmallGroup(144,154)

Series: Derived Chief Lower central Upper central

C1C3×C6 — Dic3⋊D6
C1C3C32C3×C6S3×C6C2×S32 — Dic3⋊D6
C32C3×C6 — Dic3⋊D6
C1C2C22

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 12A2B2C2D2E2F2G3A3B3C4A4B6A6B6C6D6E6F6G6H6I12A12B
 size 11266991822466224444412121212
ρ1111111111111111111111111    trivial
ρ211-1-11-1-111111-1111-1-1-1-1-11-11    linear of order 2
ρ311-1-1-111-111111111-1-1-1-1-1-111    linear of order 2
ρ41111-1-1-1-11111-111111111-1-11    linear of order 2
ρ5111-11-1-1-1111-111111111-111-1    linear of order 2
ρ611-11111-1111-1-1111-1-1-1-111-1-1    linear of order 2
ρ711-11-1-1-11111-11111-1-1-1-11-11-1    linear of order 2
ρ8111-1-1111111-1-11111111-1-1-1-1    linear of order 2
ρ9222-200002-1-1-20-12-1-1-12-11001    orthogonal lifted from D6
ρ1022-20-2000-12-1022-1-11-21101-10    orthogonal lifted from D6
ρ1122-2-200002-1-120-12-111-21100-1    orthogonal lifted from D6
ρ1222-202000-12-10-22-1-11-2110-110    orthogonal lifted from D6
ρ132-2000-22022200-2-2-200000000    orthogonal lifted from D4
ρ142-20002-2022200-2-2-200000000    orthogonal lifted from D4
ρ152220-2000-12-10-22-1-1-12-1-10110    orthogonal lifted from D6
ρ1622-2200002-1-1-20-12-111-21-1001    orthogonal lifted from D6
ρ1722202000-12-1022-1-1-12-1-10-1-10    orthogonal lifted from S3
ρ18222200002-1-120-12-1-1-12-1-100-1    orthogonal lifted from S3
ρ194-4000000-2-210022-1-30030000    orthogonal faithful
ρ204-4000000-24-200-42200000000    orthogonal lifted from S3×D4
ρ2144-400000-2-2100-2-21-122-10000    orthogonal lifted from C2×S32
ρ224-4000000-2-210022-1300-30000    orthogonal faithful
ρ234-40000004-2-2002-4200000000    orthogonal lifted from S3×D4
ρ2444400000-2-2100-2-211-2-210000    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);

Dic3⋊D6 is a maximal subgroup of
C62.2D4  C62.9D4  D6≀C2  C62⋊D4  D1223D6  D1227D6  S32×D4  Dic612D6  D1213D6  C32⋊2+ 1+4  D18⋊D6  C62⋊D6  C625D6  D64S32  (S3×C6)⋊D6  C3⋊S34D12  C6223D6  C6224D6  C6210D6
Dic3⋊D6 is a maximal quotient of
C62.10C23  C62.23C23  C62.35C23  C62.51C23  C62.53C23  D63Dic6  C62.67C23  Dic33D12  C62.82C23  C62.83C23  C62.91C23  D65D12  D12⋊D6  D12.D6  Dic6⋊D6  Dic6.D6  D12.8D6  D125D6  D12.9D6  D12.10D6  Dic6.9D6  Dic6.10D6  D12.14D6  D12.15D6  C62.95C23  C62.100C23  C62.113C23  C62.115C23  C62.116C23  C62.117C23  C62.121C23  C627D4  C628D4  C624Q8  C62.125C23  D18⋊D6  C622D6  D64S32  (S3×C6)⋊D6  C3⋊S34D12  C6223D6  C6224D6

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

1100
-1000
000-1
0011
,
0010
0001
-1000
0-100
,
-1-100
1000
0011
00-10
,
1100
0-100
0011
000-1
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

Export

Character table of Dic3⋊D6 in TeX

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