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G = D12.34D6  order 288 = 25·32

The non-split extension by D12 of D6 acting through Inn(D12)

metabelian, supersoluble, monomial

Aliases: D12.34D6, Dic6.35D6, C32⋊22- 1+4, C62.130C23, C4○D12⋊6S3, C3⋊D4.2D6, C3⋊2(Q8○D12), (S3×Dic6)⋊8C2, (C4×S3).14D6, C6.5(S3×C23), (C3×C6).5C24, D12⋊5S3⋊7C2, D6.4D6⋊3C2, (C2×C12).167D6, (S3×C6).3C23, D6.4(C22×S3), (S3×C12).31C22, (C6×C12).160C22, (C3×C12).114C23, C12.131(C22×S3), D6⋊S3.4C22, (C3×D12).43C22, C3⋊Dic3.16C23, (C3×Dic3).4C23, Dic3.3(C22×S3), (S3×Dic3).2C22, C32⋊2Q8.5C22, (C3×Dic6).44C22, C32⋊4Q8.35C22, C4.62(C2×S32), (C2×C4).36S32, C2.8(C22×S32), C22.11(C2×S32), (C3×C4○D12)⋊10C2, (C2×C6).13(C22×S3), (C2×C32⋊4Q8)⋊20C2, (C3×C3⋊D4).3C22, (C2×C3⋊Dic3).103C22, SmallGroup(288,946)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3×C6 — D12.34D6
Chief series
C1 — C3 — C32 — C3×C6 — S3×C6 — S3×Dic3 — S3×Dic6 — D12.34D6
Lower central
C32 — C3×C6 — D12.34D6
Upper central
C1 — C2 — C2×C4

Generators and relations for D12.34D6
 G = < a,b,c,d | a12=b2=1, c6=d2=a6, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=a6c5 >

Subgroups: 994 in 311 conjugacy classes, 108 normal (10 characteristic)
C1, C2, C2 [×5], C3 [×2], C3, C4 [×2], C4 [×8], C22, C22 [×4], S3 [×4], C6 [×2], C6 [×9], C2×C4, C2×C4 [×14], D4 [×10], Q8 [×10], C32, Dic3 [×4], Dic3 [×12], C12 [×4], C12 [×6], D6 [×4], C2×C6 [×2], C2×C6 [×5], C2×Q8 [×5], C4○D4 [×10], C3×S3 [×4], C3×C6, C3×C6, Dic6 [×2], Dic6 [×20], C4×S3 [×4], C4×S3 [×8], D12 [×2], C2×Dic3 [×14], C3⋊D4 [×4], C3⋊D4 [×8], C2×C12 [×2], C2×C12 [×5], C3×D4 [×6], C3×Q8 [×2], 2- 1+4, C3×Dic3 [×4], C3⋊Dic3 [×4], C3×C12 [×2], S3×C6 [×4], C62, C2×Dic6 [×7], C4○D12 [×2], C4○D12 [×4], D4⋊2S3 [×12], S3×Q8 [×4], C3×C4○D4 [×2], S3×Dic3 [×8], D6⋊S3 [×4], C32⋊2Q8 [×4], C3×Dic6 [×2], S3×C12 [×4], C3×D12 [×2], C3×C3⋊D4 [×4], C32⋊4Q8 [×4], C2×C3⋊Dic3 [×2], C6×C12, Q8○D12 [×2], S3×Dic6 [×4], D12⋊5S3 [×4], D6.4D6 [×4], C3×C4○D12 [×2], C2×C32⋊4Q8, D12.34D6
Quotients: C1, C2 [×15], C22 [×35], S3 [×2], C23 [×15], D6 [×14], C24, C22×S3 [×14], 2- 1+4, S32, S3×C23 [×2], C2×S32 [×3], Q8○D12 [×2], C22×S32, D12.34D6

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

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

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

45 conjugacy classes

class 1 2A2B2C2D2E2F3A3B3C4A4B4C4D4E4F4G4H4I4J6A6B6C···6G6H6I6J6K12A12B12C12D12E···12J12K12L12M12N
order12222223334444444444666···666661212121212···1212121212
size112666622422666618181818224···41212121222224···412121212

45 irreducible representations

dim111111222222444444
type++++++++++++-+++--
imageC1C2C2C2C2C2S3D6D6D6D6D62- 1+4S32C2×S32C2×S32Q8○D12D12.34D6
kernelD12.34D6S3×Dic6D12⋊5S3D6.4D6C3×C4○D12C2×C32⋊4Q8C4○D12Dic6C4×S3D12C3⋊D4C2×C12C32C2×C4C4C22C3C1
# reps144421224242112144

Matrix representation of D12.34D6 ►in GL4(𝔽13) generated by

7300
101000
0073
001010
,
0085
0005
5800
0800
,
61000
3300
0033
00106
,
001010
0037
61000
3300
G:=sub<GL(4,GF(13))| [7,10,0,0,3,10,0,0,0,0,7,10,0,0,3,10],[0,0,5,0,0,0,8,8,8,0,0,0,5,5,0,0],[6,3,0,0,10,3,0,0,0,0,3,10,0,0,3,6],[0,0,6,3,0,0,10,3,10,3,0,0,10,7,0,0] >;
 

D12.34D6 in GAP, Magma, Sage, TeX 

D_{12}._{34}D_6
 
% in TeX
 
G:=Group("D12.34D6");
 
// GroupNames label
 
G:=SmallGroup(288,946);
 
// by ID
 
G=gap.SmallGroup(288,946);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,120,219,100,675,80,1356,9414]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^12=b^2=1,c^6=d^2=a^6,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=a^6*c^5>;
 
// generators/relations
 

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