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## G = Dic3⋊3D12order 288 = 25·32

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

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 Dic33D12
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, Dic33D12
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, D12, C22×S3, C4⋊D4, S32, C2×D12, C4○D12, S3×D4, Q83S3, C2×S32, Dic3⋊D4, C12⋊D4, D6.6D6, S3×D12, Dic3⋊D6, Dic33D12

Smallest permutation representation of Dic33D12
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 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 4A 4B 4C 4D 4E 4F 6A ··· 6F 6G 6H 6I 6J 6K 12A ··· 12H 12I ··· 12N order 1 2 2 2 2 2 2 2 3 3 3 4 4 4 4 4 4 6 ··· 6 6 6 6 6 6 12 ··· 12 12 ··· 12 size 1 1 1 1 12 18 18 36 2 2 4 4 6 6 6 6 12 2 ··· 2 4 4 4 12 12 4 ··· 4 12 ··· 12

42 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 type + + + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 S3 S3 D4 D4 D6 D6 D6 C4○D4 D12 C4○D12 S32 S3×D4 Q8⋊3S3 C2×S32 D6.6D6 S3×D12 Dic3⋊D6 kernel Dic3⋊3D12 C6.D12 C3×Dic3⋊C4 C3×D6⋊C4 C2×C6.D6 C2×C3⋊D12 C2×C12⋊S3 Dic3⋊C4 D6⋊C4 C3×Dic3 C2×C3⋊S3 C2×Dic3 C2×C12 C22×S3 C3×C6 Dic3 C6 C2×C4 C6 C6 C22 C2 C2 C2 # reps 1 1 1 1 1 2 1 1 1 2 2 3 2 1 2 4 4 1 3 1 1 2 2 2

Matrix representation of Dic33D12 in GL8(𝔽13)

 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 1 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 7 0 0 0 0 0 0 6 3
,
 1 12 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 12 0 0 0 0 0 0 0 0 0 1 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
,
 12 0 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 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

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] >;

Dic33D12 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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