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G = D1218D6order 288 = 25·32

2nd semidirect product of D12 and D6 acting via D6/C6=C2

metabelian, supersoluble, monomial

Aliases: D1218D6, Dic616D6, C12.74D12, C62.44D4, C3⋊C83D6, C4○D124S3, C3⋊D245C2, C35(C8⋊D6), C6.66(C2×D12), (C2×C6).12D12, (C3×C12).63D4, C4.Dic37S3, C31(D4⋊D6), (C2×C12).114D6, C327(C8⋊C22), C325SD163C2, (C3×D12)⋊22C22, C12.44(C3⋊D4), (C6×C12).74C22, (C3×C12).61C23, C4.16(C3⋊D12), C12.124(C22×S3), (C3×Dic6)⋊19C22, C12⋊S3.25C22, C22.9(C3⋊D12), (C2×C4).6S32, C4.51(C2×S32), (C3×C3⋊C8)⋊3C22, C6.2(C2×C3⋊D4), (C3×C4○D12)⋊3C2, (C3×C6).65(C2×D4), C2.6(C2×C3⋊D12), (C2×C12⋊S3)⋊10C2, (C3×C4.Dic3)⋊1C2, (C2×C6).17(C3⋊D4), SmallGroup(288,473)

Series: Derived Chief Lower central Upper central

C1C3×C12 — D1218D6
C1C3C32C3×C6C3×C12C3×D12C3⋊D24 — D1218D6
C32C3×C6C3×C12 — D1218D6
C1C2C2×C4

Generators and relations for D1218D6
 G = < a,b,c,d | a12=b2=c6=d2=1, bab=dad=a-1, ac=ca, cbc-1=a6b, dbd=ab, dcd=c-1 >

Subgroups: 866 in 163 conjugacy classes, 44 normal (34 characteristic)
C1, C2, C2 [×4], C3 [×2], C3, C4 [×2], C4, C22, C22 [×5], S3 [×9], C6 [×2], C6 [×6], C8 [×2], C2×C4, C2×C4, D4 [×5], Q8, C23, C32, Dic3, C12 [×4], C12 [×3], D6 [×17], C2×C6 [×2], C2×C6 [×2], M4(2), D8 [×2], SD16 [×2], C2×D4, C4○D4, C3×S3, C3⋊S3 [×2], C3×C6, C3×C6, C3⋊C8 [×2], C24 [×2], Dic6, C4×S3, D12, D12 [×10], C3⋊D4, C2×C12 [×2], C2×C12 [×2], C3×D4 [×2], C3×Q8, C22×S3 [×4], C8⋊C22, C3×Dic3, C3×C12 [×2], S3×C6, C2×C3⋊S3 [×4], C62, C24⋊C2 [×2], D24 [×2], C4.Dic3, D4⋊S3 [×2], Q82S3 [×2], C3×M4(2), C2×D12 [×3], C4○D12, C3×C4○D4, C3×C3⋊C8 [×2], C3×Dic6, S3×C12, C3×D12, C3×C3⋊D4, C12⋊S3 [×2], C12⋊S3, C6×C12, C22×C3⋊S3, C8⋊D6, D4⋊D6, C3⋊D24 [×2], C325SD16 [×2], C3×C4.Dic3, C3×C4○D12, C2×C12⋊S3, D1218D6
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×2], C23, D6 [×6], C2×D4, D12 [×2], C3⋊D4 [×2], C22×S3 [×2], C8⋊C22, S32, C2×D12, C2×C3⋊D4, C3⋊D12 [×2], C2×S32, C8⋊D6, D4⋊D6, C2×C3⋊D12, D1218D6

Permutation representations of D1218D6
On 24 points - transitive group 24T613
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 21)(2 20)(3 19)(4 18)(5 17)(6 16)(7 15)(8 14)(9 13)(10 24)(11 23)(12 22)
(1 9 5)(2 10 6)(3 11 7)(4 12 8)(13 23 21 19 17 15)(14 24 22 20 18 16)
(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), (1,21)(2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,24)(11,23)(12,22), (1,9,5)(2,10,6)(3,11,7)(4,12,8)(13,23,21,19,17,15)(14,24,22,20,18,16), (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), (1,21)(2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,24)(11,23)(12,22), (1,9,5)(2,10,6)(3,11,7)(4,12,8)(13,23,21,19,17,15)(14,24,22,20,18,16), (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)], [(1,21),(2,20),(3,19),(4,18),(5,17),(6,16),(7,15),(8,14),(9,13),(10,24),(11,23),(12,22)], [(1,9,5),(2,10,6),(3,11,7),(4,12,8),(13,23,21,19,17,15),(14,24,22,20,18,16)], [(1,5),(2,4),(6,12),(7,11),(8,10),(13,24),(14,23),(15,22),(16,21),(17,20),(18,19)])

G:=TransitiveGroup(24,613);

39 conjugacy classes

class 1 2A2B2C2D2E3A3B3C4A4B4C6A6B6C···6G6H6I8A8B12A12B12C12D12E···12J12K12L24A24B24C24D
order122222333444666···666881212121212···12121224242424
size1121236362242212224···41212121222224···4121212121212

39 irreducible representations

dim11111122222222222244444444
type++++++++++++++++++++++++
imageC1C2C2C2C2C2S3S3D4D4D6D6D6D6D12C3⋊D4D12C3⋊D4C8⋊C22S32C3⋊D12C2×S32C3⋊D12C8⋊D6D4⋊D6D1218D6
kernelD1218D6C3⋊D24C325SD16C3×C4.Dic3C3×C4○D12C2×C12⋊S3C4.Dic3C4○D12C3×C12C62C3⋊C8Dic6D12C2×C12C12C12C2×C6C2×C6C32C2×C4C4C4C22C3C3C1
# reps12211111112112222211111224

Matrix representation of D1218D6 in GL4(𝔽73) generated by

146600
7700
001466
0077
,
006666
00597
666600
59700
,
0100
72100
00721
00720
,
17200
07200
006614
0077
G:=sub<GL(4,GF(73))| [14,7,0,0,66,7,0,0,0,0,14,7,0,0,66,7],[0,0,66,59,0,0,66,7,66,59,0,0,66,7,0,0],[0,72,0,0,1,1,0,0,0,0,72,72,0,0,1,0],[1,0,0,0,72,72,0,0,0,0,66,7,0,0,14,7] >;

D1218D6 in GAP, Magma, Sage, TeX

D_{12}\rtimes_{18}D_6
% in TeX

G:=Group("D12:18D6");
// GroupNames label

G:=SmallGroup(288,473);
// by ID

G=gap.SmallGroup(288,473);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,141,176,219,100,675,80,1356,9414]);
// Polycyclic

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

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