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## G = D12.2Dic3order 288 = 25·32

### The non-split extension by D12 of Dic3 acting through Inn(D12)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — D12.2Dic3
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — S3×C12 — S3×C3⋊C8 — D12.2Dic3
 Lower central C32 — C3×C6 — D12.2Dic3
 Upper central C1 — C4 — C2×C4

Generators and relations for D12.2Dic3
G = < a,b,c,d | a12=b2=1, c6=a6, d2=a6c3, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c5 >

Subgroups: 338 in 134 conjugacy classes, 60 normal (38 characteristic)
C1, C2, C2 [×3], C3 [×2], C3, C4 [×2], C4 [×2], C22, C22 [×2], S3 [×2], C6 [×2], C6 [×6], C8 [×4], C2×C4, C2×C4 [×2], D4 [×3], Q8, C32, Dic3 [×2], C12 [×4], C12 [×4], D6 [×2], C2×C6 [×2], C2×C6 [×3], C2×C8 [×3], M4(2) [×3], C4○D4, C3×S3 [×2], C3×C6, C3×C6, C3⋊C8 [×2], C3⋊C8 [×6], C24 [×2], Dic6, C4×S3 [×2], D12, C3⋊D4 [×2], C2×C12 [×2], C2×C12 [×3], C3×D4 [×3], C3×Q8, C8○D4, C3×Dic3 [×2], C3×C12 [×2], S3×C6 [×2], C62, S3×C8 [×2], C8⋊S3 [×2], C2×C3⋊C8, C2×C3⋊C8 [×2], C4.Dic3 [×5], C2×C24, C4○D12, C3×C4○D4, C3×C3⋊C8 [×2], C324C8 [×2], C3×Dic6, S3×C12 [×2], C3×D12, C3×C3⋊D4 [×2], C6×C12, C8○D12, D4.Dic3, S3×C3⋊C8 [×2], D6.Dic3 [×2], C6×C3⋊C8, C12.58D6, C3×C4○D12, D12.2Dic3
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3 [×2], C2×C4 [×6], C23, Dic3 [×4], D6 [×6], C22×C4, C4×S3 [×2], C2×Dic3 [×6], C22×S3 [×2], C8○D4, S32, S3×C2×C4, C22×Dic3, S3×Dic3 [×2], C2×S32, C8○D12, D4.Dic3, C2×S3×Dic3, D12.2Dic3

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

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

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

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

54 conjugacy classes

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

54 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 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 C8○D12 S32 S3×Dic3 C2×S32 S3×Dic3 D4.Dic3 D12.2Dic3 kernel D12.2Dic3 S3×C3⋊C8 D6.Dic3 C6×C3⋊C8 C12.58D6 C3×C4○D12 C3×Dic6 C3×D12 C3×C3⋊D4 C2×C3⋊C8 C4○D12 C3⋊C8 Dic6 C4×S3 D12 C3⋊D4 C2×C12 C12 C2×C6 C32 C3 C2×C4 C4 C4 C22 C3 C1 # reps 1 2 2 1 1 1 2 2 4 1 1 2 1 2 1 2 2 2 2 4 8 1 1 1 1 2 4

Matrix representation of D12.2Dic3 in GL6(𝔽73)

 1 72 0 0 0 0 1 0 0 0 0 0 0 0 46 0 0 0 0 0 46 27 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 72 0 0 0 0 0 0 27 19 0 0 0 0 27 46 0 0 0 0 0 0 72 0 0 0 0 0 0 72
,
 72 0 0 0 0 0 0 72 0 0 0 0 0 0 46 0 0 0 0 0 0 46 0 0 0 0 0 0 0 1 0 0 0 0 72 1
,
 27 0 0 0 0 0 0 27 0 0 0 0 0 0 22 0 0 0 0 0 0 22 0 0 0 0 0 0 0 46 0 0 0 0 46 0

G:=sub<GL(6,GF(73))| [1,1,0,0,0,0,72,0,0,0,0,0,0,0,46,46,0,0,0,0,0,27,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,1,0,0,0,0,0,72,0,0,0,0,0,0,27,27,0,0,0,0,19,46,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,46,0,0,0,0,0,0,46,0,0,0,0,0,0,0,72,0,0,0,0,1,1],[27,0,0,0,0,0,0,27,0,0,0,0,0,0,22,0,0,0,0,0,0,22,0,0,0,0,0,0,0,46,0,0,0,0,46,0] >;

D12.2Dic3 in GAP, Magma, Sage, TeX

D_{12}._2{\rm Dic}_3
% in TeX

G:=Group("D12.2Dic3");
// GroupNames label

G:=SmallGroup(288,462);
// by ID

G=gap.SmallGroup(288,462);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,64,422,80,1356,9414]);
// 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,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^5>;
// generators/relations

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