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G = Dic3.D12  order 288 = 25·32

4th non-split extension by Dic3 of D12 acting via D12/C12=C2

metabelian, supersoluble, monomial

Aliases: Dic3.4D12, C62.22C23, D6⋊C4⋊1S3, C6.11(S3×D4), D6⋊Dic3⋊1C2, (C4×Dic3)⋊9S3, C6.11(C2×D12), C2.16(S3×D12), (Dic3×C12)⋊1C2, (C2×C12).187D6, (C22×S3).1D6, C3⋊1(C42⋊7S3), C6.20(C4○D12), C6.D12⋊2C2, (C3×Dic3).23D4, (C2×Dic3).90D6, C32⋊2(C4.4D4), C6.34(D4⋊2S3), C6.11D12⋊14C2, (C6×C12).213C22, C2.8(D6.D6), (C6×Dic3).4C22, C3⋊1(C23.11D6), C2.11(D6.3D6), (C2×C4).41S32, (C3×D6⋊C4)⋊2C2, C22.80(C2×S32), (C3×C6).41(C2×D4), (S3×C2×C6).1C22, (C2×C32⋊2Q8)⋊2C2, (C2×C3⋊D12).2C2, (C3×C6).59(C4○D4), (C2×C6).41(C22×S3), (C22×C3⋊S3).9C22, (C2×C3⋊Dic3).22C22, SmallGroup(288,500)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C62 — Dic3.D12
Chief series
C1 — C3 — C32 — C3×C6 — C62 — S3×C2×C6 — D6⋊Dic3 — Dic3.D12
Lower central
C32 — C62 — Dic3.D12
Upper central
C1 — C22 — C2×C4

Generators and relations for Dic3.D12
 G = < a,b,c,d | a6=c12=1, b2=d2=a3, bab-1=cac-1=dad-1=a-1, bc=cb, dbd-1=a3b, dcd-1=a3c-1 >

Subgroups: 762 in 169 conjugacy classes, 48 normal (44 characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, Q8, C23, C32, Dic3, Dic3, C12, D6, C2×C6, C2×C6, C42, C22⋊C4, C2×D4, C2×Q8, C3×S3, C3⋊S3, C3×C6, Dic6, D12, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C22×S3, C22×S3, C22×C6, C4.4D4, C3×Dic3, C3×Dic3, C3⋊Dic3, C3×C12, S3×C6, C2×C3⋊S3, C62, C4×Dic3, D6⋊C4, D6⋊C4, C6.D4, C4×C12, C3×C22⋊C4, C2×Dic6, C2×D12, C2×C3⋊D4, C3⋊D12, C32⋊2Q8, C6×Dic3, C2×C3⋊Dic3, C6×C12, S3×C2×C6, C22×C3⋊S3, C42⋊7S3, C23.11D6, D6⋊Dic3, C6.D12, Dic3×C12, C3×D6⋊C4, C6.11D12, C2×C3⋊D12, C2×C32⋊2Q8, Dic3.D12
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, D12, C22×S3, C4.4D4, S32, C2×D12, C4○D12, S3×D4, D4⋊2S3, C2×S32, C42⋊7S3, C23.11D6, D6.D6, S3×D12, D6.3D6, Dic3.D12

Smallest permutation representation of Dic3.D12
►On 48 points
Generators in S48
(1 27 9 35 5 31)(2 32 6 36 10 28)(3 29 11 25 7 33)(4 34 8 26 12 30)(13 48 17 40 21 44)(14 45 22 41 18 37)(15 38 19 42 23 46)(16 47 24 43 20 39)

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

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

48 conjugacy classes

class 1 2A2B2C2D2E3A3B3C4A4B4C4D4E4F4G4H6A···6F6G6H6I6J6K12A12B12C12D12E···12J12K···12R12S12T
order122222333444444446···6666661212121212···1212···121212
size1111123622422666612362···2444121222224···46···61212

48 irreducible representations

dim111111112222222224444444
type+++++++++++++++++-++
imageC1C2C2C2C2C2C2C2S3S3D4D6D6D6C4○D4D12C4○D12S32S3×D4D4⋊2S3C2×S32D6.D6S3×D12D6.3D6
kernelDic3.D12D6⋊Dic3C6.D12Dic3×C12C3×D6⋊C4C6.11D12C2×C3⋊D12C2×C32⋊2Q8C4×Dic3D6⋊C4C3×Dic3C2×Dic3C2×C12C22×S3C3×C6Dic3C6C2×C4C6C6C22C2C2C2
# reps1111111111232144121111222

Matrix representation of Dic3.D12 ►in GL8(𝔽13)

120000000
012000000
00100000
00010000
00001000
00000100
000000012
000000112
,
08000000
80000000
00100000
00010000
00001000
00000100
00000001
00000010
,
012000000
120000000
00010000
001200000
000001200
00001100
00000001
00000010
,
50000000
08000000
000120000
001200000
000001200
000012000
00000001
00000010

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

Dic3.D12 in GAP, Magma, Sage, TeX 

{\rm Dic}_3.D_{12}
 
% in TeX
 
G:=Group("Dic3.D12");
 
// GroupNames label
 
G:=SmallGroup(288,500);
 
// by ID
 
G=gap.SmallGroup(288,500);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,141,120,422,58,1356,9414]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^6=c^12=1,b^2=d^2=a^3,b*a*b^-1=c*a*c^-1=d*a*d^-1=a^-1,b*c=c*b,d*b*d^-1=a^3*b,d*c*d^-1=a^3*c^-1>;
 
// generators/relations
 

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