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## G = Dic3⋊D4order 96 = 25·3

### 1st semidirect product of Dic3 and D4 acting via D4/C22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — Dic3⋊D4
 Chief series C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C2×C4 — Dic3⋊D4
 Lower central C3 — C2×C6 — Dic3⋊D4
 Upper central C1 — C22 — C22⋊C4

Generators and relations for Dic3⋊D4
G = < a,b,c,d | a6=c4=d2=1, b2=a3, bab-1=cac-1=dad=a-1, cbc-1=a3b, bd=db, dcd=c-1 >

Subgroups: 250 in 94 conjugacy classes, 33 normal (29 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×5], C22, C22 [×10], S3 [×3], C6 [×3], C6, C2×C4 [×2], C2×C4 [×4], D4 [×6], C23, C23 [×2], Dic3 [×2], Dic3, C12 [×2], D6 [×2], D6 [×5], C2×C6, C2×C6 [×3], C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C2×D4 [×3], C4×S3 [×2], D12 [×2], C2×Dic3 [×2], C3⋊D4 [×4], C2×C12 [×2], C22×S3 [×2], C22×C6, C4⋊D4, Dic3⋊C4, D6⋊C4, C3×C22⋊C4, S3×C2×C4, C2×D12, C2×C3⋊D4 [×2], Dic3⋊D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×4], C23, D6 [×3], C2×D4 [×2], C4○D4, C22×S3, C4⋊D4, C4○D12, S3×D4 [×2], Dic3⋊D4

Character table of Dic3⋊D4

 class 1 2A 2B 2C 2D 2E 2F 2G 3 4A 4B 4C 4D 4E 4F 6A 6B 6C 6D 6E 12A 12B 12C 12D size 1 1 1 1 4 6 6 12 2 2 2 4 6 6 12 2 2 2 4 4 4 4 4 4 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 1 -1 -1 1 1 -1 -1 -1 1 1 -1 1 1 1 1 1 -1 -1 -1 -1 linear of order 2 ρ3 1 1 1 1 -1 -1 -1 1 1 1 1 -1 -1 -1 1 1 1 1 -1 -1 1 1 -1 -1 linear of order 2 ρ4 1 1 1 1 -1 1 1 1 1 -1 -1 1 -1 -1 -1 1 1 1 -1 -1 -1 -1 1 1 linear of order 2 ρ5 1 1 1 1 -1 -1 -1 -1 1 -1 -1 1 1 1 1 1 1 1 -1 -1 -1 -1 1 1 linear of order 2 ρ6 1 1 1 1 -1 1 1 -1 1 1 1 -1 1 1 -1 1 1 1 -1 -1 1 1 -1 -1 linear of order 2 ρ7 1 1 1 1 1 1 1 -1 1 -1 -1 -1 -1 -1 1 1 1 1 1 1 -1 -1 -1 -1 linear of order 2 ρ8 1 1 1 1 1 -1 -1 -1 1 1 1 1 -1 -1 -1 1 1 1 1 1 1 1 1 1 linear of order 2 ρ9 2 2 2 2 2 0 0 0 -1 -2 -2 -2 0 0 0 -1 -1 -1 -1 -1 1 1 1 1 orthogonal lifted from D6 ρ10 2 -2 2 -2 0 0 0 0 2 0 0 0 -2 2 0 -2 2 -2 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 2 2 2 -2 0 0 0 -1 -2 -2 2 0 0 0 -1 -1 -1 1 1 1 1 -1 -1 orthogonal lifted from D6 ρ12 2 2 -2 -2 0 -2 2 0 2 0 0 0 0 0 0 2 -2 -2 0 0 0 0 0 0 orthogonal lifted from D4 ρ13 2 2 -2 -2 0 2 -2 0 2 0 0 0 0 0 0 2 -2 -2 0 0 0 0 0 0 orthogonal lifted from D4 ρ14 2 2 2 2 2 0 0 0 -1 2 2 2 0 0 0 -1 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ15 2 2 2 2 -2 0 0 0 -1 2 2 -2 0 0 0 -1 -1 -1 1 1 -1 -1 1 1 orthogonal lifted from D6 ρ16 2 -2 2 -2 0 0 0 0 2 0 0 0 2 -2 0 -2 2 -2 0 0 0 0 0 0 orthogonal lifted from D4 ρ17 2 -2 -2 2 0 0 0 0 2 -2i 2i 0 0 0 0 -2 -2 2 0 0 -2i 2i 0 0 complex lifted from C4○D4 ρ18 2 -2 -2 2 0 0 0 0 2 2i -2i 0 0 0 0 -2 -2 2 0 0 2i -2i 0 0 complex lifted from C4○D4 ρ19 2 -2 -2 2 0 0 0 0 -1 -2i 2i 0 0 0 0 1 1 -1 √-3 -√-3 i -i √3 -√3 complex lifted from C4○D12 ρ20 2 -2 -2 2 0 0 0 0 -1 2i -2i 0 0 0 0 1 1 -1 -√-3 √-3 -i i √3 -√3 complex lifted from C4○D12 ρ21 2 -2 -2 2 0 0 0 0 -1 -2i 2i 0 0 0 0 1 1 -1 -√-3 √-3 i -i -√3 √3 complex lifted from C4○D12 ρ22 2 -2 -2 2 0 0 0 0 -1 2i -2i 0 0 0 0 1 1 -1 √-3 -√-3 -i i -√3 √3 complex lifted from C4○D12 ρ23 4 4 -4 -4 0 0 0 0 -2 0 0 0 0 0 0 -2 2 2 0 0 0 0 0 0 orthogonal lifted from S3×D4 ρ24 4 -4 4 -4 0 0 0 0 -2 0 0 0 0 0 0 2 -2 2 0 0 0 0 0 0 orthogonal lifted from S3×D4

Smallest permutation representation of Dic3⋊D4
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 28 4 25)(2 27 5 30)(3 26 6 29)(7 34 10 31)(8 33 11 36)(9 32 12 35)(13 21 16 24)(14 20 17 23)(15 19 18 22)(37 45 40 48)(38 44 41 47)(39 43 42 46)
(1 36 13 37)(2 35 14 42)(3 34 15 41)(4 33 16 40)(5 32 17 39)(6 31 18 38)(7 19 44 26)(8 24 45 25)(9 23 46 30)(10 22 47 29)(11 21 48 28)(12 20 43 27)
(1 8)(2 7)(3 12)(4 11)(5 10)(6 9)(13 45)(14 44)(15 43)(16 48)(17 47)(18 46)(19 42)(20 41)(21 40)(22 39)(23 38)(24 37)(25 36)(26 35)(27 34)(28 33)(29 32)(30 31)

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

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

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

Matrix representation of Dic3⋊D4 in GL6(𝔽13)

 0 1 0 0 0 0 12 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12
,
 12 0 0 0 0 0 1 1 0 0 0 0 0 0 7 10 0 0 0 0 8 6 0 0 0 0 0 0 12 3 0 0 0 0 8 1
,
 1 0 0 0 0 0 12 12 0 0 0 0 0 0 5 0 0 0 0 0 6 8 0 0 0 0 0 0 12 3 0 0 0 0 0 1
,
 12 0 0 0 0 0 1 1 0 0 0 0 0 0 9 11 0 0 0 0 1 4 0 0 0 0 0 0 1 0 0 0 0 0 0 1

G:=sub<GL(6,GF(13))| [0,12,0,0,0,0,1,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[12,1,0,0,0,0,0,1,0,0,0,0,0,0,7,8,0,0,0,0,10,6,0,0,0,0,0,0,12,8,0,0,0,0,3,1],[1,12,0,0,0,0,0,12,0,0,0,0,0,0,5,6,0,0,0,0,0,8,0,0,0,0,0,0,12,0,0,0,0,0,3,1],[12,1,0,0,0,0,0,1,0,0,0,0,0,0,9,1,0,0,0,0,11,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

Dic3⋊D4 in GAP, Magma, Sage, TeX

{\rm Dic}_3\rtimes D_4
% in TeX

G:=Group("Dic3:D4");
// GroupNames label

G:=SmallGroup(96,91);
// by ID

G=gap.SmallGroup(96,91);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,55,506,188,2309]);
// Polycyclic

G:=Group<a,b,c,d|a^6=c^4=d^2=1,b^2=a^3,b*a*b^-1=c*a*c^-1=d*a*d=a^-1,c*b*c^-1=a^3*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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