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G = D12⋊D6order 288 = 25·32

4th semidirect product of D12 and D6 acting via D6/C3=C22

metabelian, supersoluble, monomial

Aliases: D124D6, D43S32, C3⋊S33D8, C3⋊C813D6, C33(S3×D8), D4⋊S32S3, (C3×D4)⋊1D6, C328(C2×D8), C6.55(S3×D4), D6⋊D63C2, C3⋊D248C2, (C3×D12)⋊6C22, C3⋊Dic3.19D4, (C3×C12).3C23, C12.3(C22×S3), C12⋊S34C22, C12.29D63C2, (D4×C32)⋊3C22, C2.15(Dic3⋊D6), C4.3(C2×S32), (D4×C3⋊S3)⋊1C2, (C3×D4⋊S3)⋊2C2, (C3×C3⋊C8)⋊6C22, (C2×C3⋊S3).56D4, (C3×C6).118(C2×D4), (C4×C3⋊S3).11C22, SmallGroup(288,574)

Series: Derived Chief Lower central Upper central

C1C3×C12 — D12⋊D6
C1C3C32C3×C6C3×C12C3×D12D6⋊D6 — D12⋊D6
C32C3×C6C3×C12 — D12⋊D6
C1C2C4D4

Generators and relations for D12⋊D6
 G = < a,b,c,d | a12=b2=c6=d2=1, bab=dad=a-1, cac-1=a7, cbc-1=a3b, dbd=a7b, dcd=c-1 >

Subgroups: 1026 in 179 conjugacy classes, 40 normal (16 characteristic)
C1, C2, C2 [×6], C3 [×2], C3, C4, C4, C22 [×9], S3 [×12], C6 [×2], C6 [×7], C8 [×2], C2×C4, D4, D4 [×5], C23 [×2], C32, Dic3 [×3], C12 [×2], C12, D6 [×21], C2×C6 [×6], C2×C8, D8 [×4], C2×D4 [×2], C3×S3 [×2], C3⋊S3 [×2], C3⋊S3, C3×C6, C3×C6, C3⋊C8 [×2], C24 [×2], C4×S3 [×3], D12 [×2], D12 [×3], C3⋊D4 [×6], C3×D4 [×2], C3×D4 [×3], C22×S3 [×6], C2×D8, C3⋊Dic3, C3×C12, S32 [×2], S3×C6 [×2], C2×C3⋊S3, C2×C3⋊S3 [×3], C62, S3×C8 [×2], D24 [×2], D4⋊S3 [×2], D4⋊S3 [×2], C3×D8 [×2], S3×D4 [×5], C3×C3⋊C8 [×2], D6⋊S3, C3×D12 [×2], C4×C3⋊S3, C12⋊S3, C327D4, D4×C32, C2×S32, C22×C3⋊S3, S3×D8 [×2], C12.29D6, C3⋊D24 [×2], C3×D4⋊S3 [×2], D6⋊D6, D4×C3⋊S3, D12⋊D6
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×2], C23, D6 [×6], D8 [×2], C2×D4, C22×S3 [×2], C2×D8, S32, S3×D4 [×2], C2×S32, S3×D8 [×2], Dic3⋊D6, D12⋊D6

Permutation representations of D12⋊D6
On 24 points - transitive group 24T668
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 24)(2 23)(3 22)(4 21)(5 20)(6 19)(7 18)(8 17)(9 16)(10 15)(11 14)(12 13)
(1 9 5)(2 4 6 8 10 12)(3 11 7)(13 20 21 16 17 24)(14 15 22 23 18 19)
(1 5)(2 4)(6 12)(7 11)(8 10)(13 24)(14 23)(15 22)(16 21)(17 20)(18 19)

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

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

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

G:=TransitiveGroup(24,668);

33 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B3C4A4B6A6B6C6D6E6F6G6H6I8A8B8C8D12A12B12C24A24B24C24D
order1222222233344666666666888812121224242424
size1149912123622421822488882424666644812121212

33 irreducible representations

dim1111112222222444448
type+++++++++++++++++++
imageC1C2C2C2C2C2S3D4D4D6D6D6D8S32S3×D4C2×S32S3×D8Dic3⋊D6D12⋊D6
kernelD12⋊D6C12.29D6C3⋊D24C3×D4⋊S3D6⋊D6D4×C3⋊S3D4⋊S3C3⋊Dic3C2×C3⋊S3C3⋊C8D12C3×D4C3⋊S3D4C6C4C3C2C1
# reps1122112112224121421

Matrix representation of D12⋊D6 in GL6(𝔽73)

010000
7200000
0072100
0072000
000010
000001
,
16160000
16570000
001000
0017200
000010
000001
,
100000
0720000
0072000
0007200
000001
00007272
,
100000
0720000
0007200
0072000
000001
000010

G:=sub<GL(6,GF(73))| [0,72,0,0,0,0,1,0,0,0,0,0,0,0,72,72,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[16,16,0,0,0,0,16,57,0,0,0,0,0,0,1,1,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,1,72],[1,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,72,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

D12⋊D6 in GAP, Magma, Sage, TeX

D_{12}\rtimes D_6
% in TeX

G:=Group("D12:D6");
// GroupNames label

G:=SmallGroup(288,574);
// by ID

G=gap.SmallGroup(288,574);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,254,135,675,346,185,80,1356,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^12=b^2=c^6=d^2=1,b*a*b=d*a*d=a^-1,c*a*c^-1=a^7,c*b*c^-1=a^3*b,d*b*d=a^7*b,d*c*d=c^-1>;
// generators/relations

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