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G = D6.3D12order 288 = 25·32

3rd non-split extension by D6 of D12 acting via D12/C12=C2

metabelian, supersoluble, monomial

Aliases: D6.3D12, C24.51D6, Dic127S3, Dic6.18D6, Dic3.14D12, C8.10S32, (S3×C8)⋊3S3, C3⋊C8.26D6, (S3×C24)⋊2C2, C6.10(S3×D4), C325D87C2, C33(C4○D24), (C4×S3).38D6, (S3×C6).22D4, C6.10(C2×D12), C2.15(S3×D12), C325(C4○D8), D6.6D63C2, C31(D24⋊C2), (C3×Dic12)⋊10C2, C325SD166C2, (C3×C12).53C23, C12.73(C22×S3), (C3×C24).24C22, (C3×Dic3).27D4, (S3×C12).46C22, C12⋊S3.5C22, (C3×Dic6).6C22, C4.49(C2×S32), (C3×C6).37(C2×D4), (C3×C3⋊C8).33C22, SmallGroup(288,456)

Series: Derived Chief Lower central Upper central

C1C3×C12 — D6.3D12
C1C3C32C3×C6C3×C12S3×C12D6.6D6 — D6.3D12
C32C3×C6C3×C12 — D6.3D12
C1C2C4C8

Generators and relations for D6.3D12
 G = < a,b,c,d | a6=b2=1, c12=d2=a3, bab=a-1, ac=ca, ad=da, bc=cb, dbd-1=a3b, dcd-1=a3c11 >

Subgroups: 674 in 135 conjugacy classes, 40 normal (30 characteristic)
C1, C2, C2 [×3], C3 [×2], C3, C4, C4 [×3], C22 [×3], S3 [×7], C6 [×2], C6 [×2], C8, C8, C2×C4 [×3], D4 [×4], Q8 [×2], C32, Dic3, Dic3 [×2], C12 [×2], C12 [×4], D6, D6 [×6], C2×C6, C2×C8, D8, SD16 [×2], Q16, C4○D4 [×2], C3×S3, C3⋊S3 [×2], C3×C6, C3⋊C8, C24 [×2], C24 [×2], Dic6 [×2], C4×S3, C4×S3 [×4], D12 [×8], C3⋊D4 [×2], C2×C12, C3×Q8 [×2], C4○D8, C3×Dic3, C3×Dic3 [×2], C3×C12, S3×C6, C2×C3⋊S3 [×2], S3×C8, C24⋊C2 [×2], D24 [×3], Dic12, Q82S3 [×2], C2×C24, C3×Q16, C4○D12 [×2], Q83S3 [×2], C3×C3⋊C8, C3×C24, C6.D6 [×2], C3⋊D12 [×2], C3×Dic6 [×2], S3×C12, C12⋊S3 [×2], C4○D24, D24⋊C2, C325SD16 [×2], S3×C24, C3×Dic12, C325D8, D6.6D6 [×2], D6.3D12
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×2], C23, D6 [×6], C2×D4, D12 [×2], C22×S3 [×2], C4○D8, S32, C2×D12, S3×D4, C2×S32, C4○D24, D24⋊C2, S3×D12, D6.3D12

Smallest permutation representation of D6.3D12
On 48 points
Generators in S48
(1 21 17 13 9 5)(2 22 18 14 10 6)(3 23 19 15 11 7)(4 24 20 16 12 8)(25 29 33 37 41 45)(26 30 34 38 42 46)(27 31 35 39 43 47)(28 32 36 40 44 48)
(1 45)(2 46)(3 47)(4 48)(5 25)(6 26)(7 27)(8 28)(9 29)(10 30)(11 31)(12 32)(13 33)(14 34)(15 35)(16 36)(17 37)(18 38)(19 39)(20 40)(21 41)(22 42)(23 43)(24 44)
(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 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48)
(1 36 13 48)(2 35 14 47)(3 34 15 46)(4 33 16 45)(5 32 17 44)(6 31 18 43)(7 30 19 42)(8 29 20 41)(9 28 21 40)(10 27 22 39)(11 26 23 38)(12 25 24 37)

G:=sub<Sym(48)| (1,21,17,13,9,5)(2,22,18,14,10,6)(3,23,19,15,11,7)(4,24,20,16,12,8)(25,29,33,37,41,45)(26,30,34,38,42,46)(27,31,35,39,43,47)(28,32,36,40,44,48), (1,45)(2,46)(3,47)(4,48)(5,25)(6,26)(7,27)(8,28)(9,29)(10,30)(11,31)(12,32)(13,33)(14,34)(15,35)(16,36)(17,37)(18,38)(19,39)(20,40)(21,41)(22,42)(23,43)(24,44), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48), (1,36,13,48)(2,35,14,47)(3,34,15,46)(4,33,16,45)(5,32,17,44)(6,31,18,43)(7,30,19,42)(8,29,20,41)(9,28,21,40)(10,27,22,39)(11,26,23,38)(12,25,24,37)>;

G:=Group( (1,21,17,13,9,5)(2,22,18,14,10,6)(3,23,19,15,11,7)(4,24,20,16,12,8)(25,29,33,37,41,45)(26,30,34,38,42,46)(27,31,35,39,43,47)(28,32,36,40,44,48), (1,45)(2,46)(3,47)(4,48)(5,25)(6,26)(7,27)(8,28)(9,29)(10,30)(11,31)(12,32)(13,33)(14,34)(15,35)(16,36)(17,37)(18,38)(19,39)(20,40)(21,41)(22,42)(23,43)(24,44), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48), (1,36,13,48)(2,35,14,47)(3,34,15,46)(4,33,16,45)(5,32,17,44)(6,31,18,43)(7,30,19,42)(8,29,20,41)(9,28,21,40)(10,27,22,39)(11,26,23,38)(12,25,24,37) );

G=PermutationGroup([(1,21,17,13,9,5),(2,22,18,14,10,6),(3,23,19,15,11,7),(4,24,20,16,12,8),(25,29,33,37,41,45),(26,30,34,38,42,46),(27,31,35,39,43,47),(28,32,36,40,44,48)], [(1,45),(2,46),(3,47),(4,48),(5,25),(6,26),(7,27),(8,28),(9,29),(10,30),(11,31),(12,32),(13,33),(14,34),(15,35),(16,36),(17,37),(18,38),(19,39),(20,40),(21,41),(22,42),(23,43),(24,44)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48)], [(1,36,13,48),(2,35,14,47),(3,34,15,46),(4,33,16,45),(5,32,17,44),(6,31,18,43),(7,30,19,42),(8,29,20,41),(9,28,21,40),(10,27,22,39),(11,26,23,38),(12,25,24,37)])

45 conjugacy classes

class 1 2A2B2C2D3A3B3C4A4B4C4D4E6A6B6C6D6E8A8B8C8D12A12B12C12D12E12F12G12H12I24A24B24C24D24E···24J24K24L24M24N
order12222333444446666688881212121212121212122424242424···2424242424
size116363622423312122246622662244466242422224···46666

45 irreducible representations

dim111111222222222222444444
type++++++++++++++++++++++
imageC1C2C2C2C2C2S3S3D4D4D6D6D6D6D12D12C4○D8C4○D24S32S3×D4C2×S32D24⋊C2S3×D12D6.3D12
kernelD6.3D12C325SD16S3×C24C3×Dic12C325D8D6.6D6S3×C8Dic12C3×Dic3S3×C6C3⋊C8C24Dic6C4×S3Dic3D6C32C3C8C6C4C3C2C1
# reps121112111112212248111224

Matrix representation of D6.3D12 in GL6(𝔽73)

7200000
0720000
0072100
0072000
000010
000001
,
0270000
4600000
001000
0017200
000010
000001
,
57160000
57570000
001000
000100
0000721
0000720
,
67670000
6760000
0072000
0007200
0000720
0000721

G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,72,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,46,0,0,0,0,27,0,0,0,0,0,0,0,1,1,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[57,57,0,0,0,0,16,57,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,1,0],[67,67,0,0,0,0,67,6,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,72,0,0,0,0,0,1] >;

D6.3D12 in GAP, Magma, Sage, TeX

D_6._3D_{12}
% in TeX

G:=Group("D6.3D12");
// GroupNames label

G:=SmallGroup(288,456);
// by ID

G=gap.SmallGroup(288,456);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,120,135,142,346,80,1356,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^6=b^2=1,c^12=d^2=a^3,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=a^3*b,d*c*d^-1=a^3*c^11>;
// generators/relations

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