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G = D6.D12  order 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⋊C4⋊16S3, C4⋊Dic3⋊6S3, (S3×C6).6D4, C6.14(S3×D4), C6.15(C2×D12), C2.18(S3×D12), (C2×C12).228D6, Dic3⋊Dic3⋊5C2, C6.54(C4○D12), C3⋊4(D6.D4), C6.D12⋊7C2, (C2×Dic3).24D6, (C22×S3).37D6, C6.11D12⋊10C2, C6.41(D4⋊2S3), (C6×C12).184C22, C6.34(Q8⋊3S3), C2.17(D12⋊S3), C2.17(D6.3D6), C3⋊1(C23.21D6), (C6×Dic3).13C22, C32⋊6(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

Derived series
C1 — C62 — D6.D12
Chief series
C1 — C3 — C32 — C3×C6 — C62 — S3×C2×C6 — C2×S3×Dic3 — D6.D12
Lower central
C32 — C62 — D6.D12
Upper central
C1 — C22 — C2×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, C2, C3, C3, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, C23, C32, Dic3, C12, D6, D6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C3×S3, C3⋊S3, C3×C6, C4×S3, D12, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C22×S3, C22×S3, C22×C6, C22.D4, C3×Dic3, C3⋊Dic3, C3×C12, S3×C6, S3×C6, C2×C3⋊S3, C62, Dic3⋊C4, C4⋊Dic3, C4⋊Dic3, D6⋊C4, D6⋊C4, C3×C22⋊C4, C3×C4⋊C4, S3×C2×C4, C2×D12, C22×Dic3, C2×C3⋊D4, S3×Dic3, C3⋊D12, C6×Dic3, 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, C22, S3, D4, C23, D6, C2×D4, C4○D4, D12, C22×S3, C22.D4, S32, C2×D12, C4○D12, S3×D4, D4⋊2S3, Q8⋊3S3, 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 26 9 34 5 30)(2 27 10 35 6 31)(3 28 11 36 7 32)(4 29 12 25 8 33)(13 37 17 41 21 45)(14 38 18 42 22 46)(15 39 19 43 23 47)(16 40 20 44 24 48)

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

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

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×D4D4⋊2S3Q8⋊3S3C2×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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