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G = D6.D12order 288 = 25·32

5th non-split extension by D6 of D12 acting via D12/D6=C2

metabelian, supersoluble, monomial

Aliases: D6.8D12, C62.60C23, D6⋊C416S3, C4⋊Dic36S3, (S3×C6).6D4, C6.14(S3×D4), C6.15(C2×D12), C2.18(S3×D12), (C2×C12).228D6, Dic3⋊Dic35C2, C6.54(C4○D12), C34(D6.D4), C6.D127C2, (C2×Dic3).24D6, (C22×S3).37D6, C6.11D1210C2, C6.41(D42S3), (C6×C12).184C22, C6.34(Q83S3), C2.17(D12⋊S3), C2.17(D6.3D6), C31(C23.21D6), (C6×Dic3).13C22, C326(C22.D4), (C2×C4).25S32, (C3×D6⋊C4)⋊14C2, (C2×S3×Dic3)⋊12C2, (C3×C6).47(C2×D4), (C3×C4⋊Dic3)⋊12C2, C22.106(C2×S32), (S3×C2×C6).19C22, (C2×C3⋊D12).6C2, (C3×C6).36(C4○D4), (C2×C6).79(C22×S3), (C22×C3⋊S3).16C22, (C2×C3⋊Dic3).44C22, SmallGroup(288,538)

Series: Derived Chief Lower central Upper central

C1C62 — D6.D12
C1C3C32C3×C6C62S3×C2×C6C2×S3×Dic3 — D6.D12
C32C62 — D6.D12
C1C22C2×C4

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

Subgroups: 762 in 173 conjugacy classes, 48 normal (44 characteristic)
C1, C2 [×3], C2 [×3], C3 [×2], C3, C4 [×5], C22, C22 [×7], S3 [×6], C6 [×6], C6 [×5], C2×C4, C2×C4 [×6], D4 [×2], C23 [×2], C32, Dic3 [×6], C12 [×6], D6 [×2], D6 [×12], C2×C6 [×2], C2×C6 [×5], C22⋊C4 [×3], C4⋊C4 [×2], C22×C4, C2×D4, C3×S3 [×2], C3⋊S3, C3×C6 [×3], C4×S3 [×2], D12 [×2], C2×Dic3 [×3], C2×Dic3 [×5], C3⋊D4 [×2], C2×C12 [×2], C2×C12 [×4], C22×S3, C22×S3 [×3], C22×C6, C22.D4, C3×Dic3 [×3], C3⋊Dic3, C3×C12, S3×C6 [×2], S3×C6 [×2], C2×C3⋊S3 [×3], C62, Dic3⋊C4, C4⋊Dic3, C4⋊Dic3, D6⋊C4, D6⋊C4 [×5], C3×C22⋊C4, C3×C4⋊C4, S3×C2×C4, C2×D12, C22×Dic3, C2×C3⋊D4, S3×Dic3 [×2], C3⋊D12 [×2], C6×Dic3 [×3], C2×C3⋊Dic3, C6×C12, S3×C2×C6, C22×C3⋊S3, C23.21D6, D6.D4, C6.D12, Dic3⋊Dic3, C3×C4⋊Dic3, C3×D6⋊C4, C6.11D12, C2×S3×Dic3, C2×C3⋊D12, D6.D12
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×2], C23, D6 [×6], C2×D4, C4○D4 [×2], D12 [×2], C22×S3 [×2], C22.D4, S32, C2×D12, C4○D12, S3×D4, D42S3 [×2], Q83S3, C2×S32, C23.21D6, D6.D4, D12⋊S3, S3×D12, D6.3D6, D6.D12

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

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

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

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

42 conjugacy classes

class 1 2A2B2C2D2E2F3A3B3C4A4B4C4D4E4F4G6A···6F6G6H6I6J6K12A···12H12I···12N
order122222233344444446···66666612···1212···12
size11116636224466121218182···244412124···412···12

42 irreducible representations

dim1111111122222222244444444
type+++++++++++++++++-+++
imageC1C2C2C2C2C2C2C2S3S3D4D6D6D6C4○D4D12C4○D12S32S3×D4D42S3Q83S3C2×S32D12⋊S3S3×D12D6.3D6
kernelD6.D12C6.D12Dic3⋊Dic3C3×C4⋊Dic3C3×D6⋊C4C6.11D12C2×S3×Dic3C2×C3⋊D12C4⋊Dic3D6⋊C4S3×C6C2×Dic3C2×C12C22×S3C3×C6D6C6C2×C4C6C6C6C22C2C2C2
# reps1111111111232144411211222

Matrix representation of D6.D12 in GL8(𝔽13)

120000000
012000000
00100000
00010000
000012100
000012000
00000010
00000001
,
512000000
118000000
00100000
00010000
00001000
000011200
00000010
00000001
,
80000000
25000000
001220000
001210000
000012000
000001200
000000121
000000120
,
81000000
05000000
001200000
001210000
000012000
000001200
000000120
000000121

G:=sub<GL(8,GF(13))| [12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[5,11,0,0,0,0,0,0,12,8,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[8,2,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,2,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,1,0],[8,0,0,0,0,0,0,0,1,5,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,1] >;

D6.D12 in GAP, Magma, Sage, TeX

D_6.D_{12}
% in TeX

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

G:=SmallGroup(288,538);
// by ID

G=gap.SmallGroup(288,538);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,64,254,219,100,1356,9414]);
// Polycyclic

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

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