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G = Dic3⋊3D12  order 288 = 25·32

2nd semidirect product of Dic3 and D12 acting via D12/D6=C2

metabelian, supersoluble, monomial

Aliases: Dic3⋊3D12, C62.80C23, D6⋊C4⋊5S3, C6.23(S3×D4), Dic3⋊C4⋊8S3, (C3×Dic3)⋊2D4, (C2×C12).27D6, C2.25(S3×D12), C6.24(C2×D12), C3⋊1(Dic3⋊D4), C3⋊1(C12⋊D4), C32⋊8(C4⋊D4), (C6×C12).8C22, C6.17(C4○D12), C6.D12⋊8C2, (C2×Dic3).76D6, (C22×S3).15D6, C2.11(Dic3⋊D6), C6.18(Q8⋊3S3), C2.19(D6.6D6), (C6×Dic3).69C22, (C2×C4).33S32, (C2×C3⋊S3)⋊1D4, (C3×D6⋊C4)⋊7C2, (C2×C12⋊S3)⋊1C2, C22.118(C2×S32), (C2×C6.D6)⋊1C2, (C3×C6).106(C2×D4), (S3×C2×C6).30C22, (C2×C3⋊D12)⋊16C2, (C3×Dic3⋊C4)⋊22C2, (C3×C6).50(C4○D4), (C2×C6).99(C22×S3), (C22×C3⋊S3).24C22, SmallGroup(288,558)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C62 — Dic3⋊3D12
Chief series
C1 — C3 — C32 — C3×C6 — C62 — S3×C2×C6 — C2×C3⋊D12 — Dic3⋊3D12
Lower central
C32 — C62 — Dic3⋊3D12
Upper central
C1 — C22 — C2×C4

Generators and relations for Dic3⋊3D12
 G = < a,b,c,d | a6=c12=d2=1, b2=a3, bab-1=dad=a-1, ac=ca, cbc-1=a3b, bd=db, dcd=c-1 >

Subgroups: 1090 in 215 conjugacy classes, 50 normal (44 characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, C23, C32, Dic3, Dic3, C12, D6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C3×S3, C3⋊S3, C3×C6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C22×S3, C22×S3, C22×C6, C4⋊D4, C3×Dic3, C3×Dic3, C3×C12, S3×C6, C2×C3⋊S3, C2×C3⋊S3, C62, Dic3⋊C4, D6⋊C4, D6⋊C4, C3×C22⋊C4, C3×C4⋊C4, S3×C2×C4, C2×D12, C2×C3⋊D4, C6.D6, C3⋊D12, C6×Dic3, C12⋊S3, C6×C12, S3×C2×C6, C22×C3⋊S3, Dic3⋊D4, C12⋊D4, C6.D12, C3×Dic3⋊C4, C3×D6⋊C4, C2×C6.D6, C2×C3⋊D12, C2×C12⋊S3, Dic3⋊3D12
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, D12, C22×S3, C4⋊D4, S32, C2×D12, C4○D12, S3×D4, Q8⋊3S3, C2×S32, Dic3⋊D4, C12⋊D4, D6.6D6, S3×D12, Dic3⋊D6, Dic3⋊3D12

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

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

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

42 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B3C4A4B4C4D4E4F6A···6F6G6H6I6J6K12A···12H12I···12N
order122222223334444446···66666612···1212···12
size11111218183622446666122···244412124···412···12

42 irreducible representations

dim111111122222222224444444
type++++++++++++++++++++++
imageC1C2C2C2C2C2C2S3S3D4D4D6D6D6C4○D4D12C4○D12S32S3×D4Q8⋊3S3C2×S32D6.6D6S3×D12Dic3⋊D6
kernelDic3⋊3D12C6.D12C3×Dic3⋊C4C3×D6⋊C4C2×C6.D6C2×C3⋊D12C2×C12⋊S3Dic3⋊C4D6⋊C4C3×Dic3C2×C3⋊S3C2×Dic3C2×C12C22×S3C3×C6Dic3C6C2×C4C6C6C22C2C2C2
# reps111112111223212441311222

Matrix representation of Dic3⋊3D12 ►in GL8(𝔽13)

10000000
01000000
000120000
001120000
00001000
00000100
000000120
000000012
,
120000000
012000000
000120000
001200000
000012000
000001200
000000107
00000063
,
112000000
10000000
001200000
000120000
00000100
000012000
00000001
00000010
,
120000000
121000000
00010000
00100000
000012000
00000100
00000010
00000001

G:=sub<GL(8,GF(13))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12],[12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,10,6,0,0,0,0,0,0,7,3],[1,1,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[12,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1] >;
 

Dic3⋊3D12 in GAP, Magma, Sage, TeX 

{\rm Dic}_3\rtimes_3D_{12}
 
% in TeX
 
G:=Group("Dic3:3D12");
 
// GroupNames label
 
G:=SmallGroup(288,558);
 
// by ID
 
G=gap.SmallGroup(288,558);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,141,64,422,135,142,1356,9414]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^6=c^12=d^2=1,b^2=a^3,b*a*b^-1=d*a*d=a^-1,a*c=c*a,c*b*c^-1=a^3*b,b*d=d*b,d*c*d=c^-1>;
 
// generators/relations
 

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