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## G = D12.Dic3order 288 = 25·32

### The non-split extension by D12 of Dic3 acting via Dic3/C6=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — D12.Dic3
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — S3×C12 — S3×C3⋊C8 — D12.Dic3
 Lower central C32 — C3×C6 — D12.Dic3
 Upper central C1 — C4 — C2×C4

Generators and relations for D12.Dic3
G = < a,b,c,d | a12=b2=1, c6=a6, d2=a6c3, bab=a-1, ac=ca, dad-1=a7, bc=cb, dbd-1=a6b, dcd-1=c5 >

Subgroups: 338 in 135 conjugacy classes, 60 normal (36 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, Q8, C32, Dic3, C12, C12, D6, C2×C6, C2×C6, C2×C8, M4(2), C4○D4, C3×S3, C3×C6, C3×C6, C3⋊C8, C3⋊C8, C24, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×Q8, C8○D4, C3×Dic3, C3×C12, S3×C6, C62, S3×C8, C8⋊S3, C2×C3⋊C8, C4.Dic3, C4.Dic3, C3×M4(2), C4○D12, C3×C4○D4, C3×C3⋊C8, C324C8, C3×Dic6, S3×C12, C3×D12, C3×C3⋊D4, C6×C12, D12.C4, D4.Dic3, S3×C3⋊C8, D6.Dic3, C3×C4.Dic3, C2×C324C8, C3×C4○D12, D12.Dic3
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, Dic3, D6, C22×C4, C4×S3, C2×Dic3, C22×S3, C8○D4, S32, S3×C2×C4, C22×Dic3, S3×Dic3, C2×S32, D12.C4, D4.Dic3, C2×S3×Dic3, D12.Dic3

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

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

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

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

48 conjugacy classes

 class 1 2A 2B 2C 2D 3A 3B 3C 4A 4B 4C 4D 4E 6A 6B 6C ··· 6G 6H 6I 8A 8B 8C 8D 8E 8F 8G 8H 8I 8J 12A 12B 12C 12D 12E ··· 12J 12K 12L 24A 24B 24C 24D order 1 2 2 2 2 3 3 3 4 4 4 4 4 6 6 6 ··· 6 6 6 8 8 8 8 8 8 8 8 8 8 12 12 12 12 12 ··· 12 12 12 24 24 24 24 size 1 1 2 6 6 2 2 4 1 1 2 6 6 2 2 4 ··· 4 12 12 6 6 6 6 9 9 9 9 18 18 2 2 2 2 4 ··· 4 12 12 12 12 12 12

48 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 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 C4 C4 C4 S3 S3 D6 Dic3 D6 Dic3 Dic3 D6 C4×S3 C4×S3 C8○D4 S32 S3×Dic3 C2×S32 S3×Dic3 D12.C4 D4.Dic3 D12.Dic3 kernel D12.Dic3 S3×C3⋊C8 D6.Dic3 C3×C4.Dic3 C2×C32⋊4C8 C3×C4○D12 C3×Dic6 C3×D12 C3×C3⋊D4 C4.Dic3 C4○D12 C3⋊C8 Dic6 C4×S3 D12 C3⋊D4 C2×C12 C12 C2×C6 C32 C2×C4 C4 C4 C22 C3 C3 C1 # reps 1 2 2 1 1 1 2 2 4 1 1 2 1 2 1 2 2 2 2 4 1 1 1 1 2 2 4

Matrix representation of D12.Dic3 in GL4(𝔽5) generated by

 0 1 1 4 2 3 2 3 2 2 0 1 1 1 2 2
,
 2 4 0 2 2 3 1 0 0 4 2 1 2 0 3 3
,
 4 0 1 0 0 2 0 3 2 0 4 0 0 1 0 1
,
 0 3 0 4 1 0 2 0 0 2 0 3 1 0 1 0
G:=sub<GL(4,GF(5))| [0,2,2,1,1,3,2,1,1,2,0,2,4,3,1,2],[2,2,0,2,4,3,4,0,0,1,2,3,2,0,1,3],[4,0,2,0,0,2,0,1,1,0,4,0,0,3,0,1],[0,1,0,1,3,0,2,0,0,2,0,1,4,0,3,0] >;

D12.Dic3 in GAP, Magma, Sage, TeX

D_{12}.{\rm Dic}_3
% in TeX

G:=Group("D12.Dic3");
// GroupNames label

G:=SmallGroup(288,463);
// by ID

G=gap.SmallGroup(288,463);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,64,422,219,80,1356,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^12=b^2=1,c^6=a^6,d^2=a^6*c^3,b*a*b=a^-1,a*c=c*a,d*a*d^-1=a^7,b*c=c*b,d*b*d^-1=a^6*b,d*c*d^-1=c^5>;
// generators/relations

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