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## G = D4×Dic3order 96 = 25·3

### Direct product of D4 and Dic3

Series: Derived Chief Lower central Upper central

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

Generators and relations for D4×Dic3
G = < a,b,c,d | a4=b2=c6=1, d2=c3, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 178 in 94 conjugacy classes, 51 normal (19 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×2], C4 [×5], C22, C22 [×4], C22 [×4], C6 [×3], C6 [×4], C2×C4, C2×C4 [×8], D4 [×4], C23 [×2], Dic3 [×2], Dic3 [×3], C12 [×2], C2×C6, C2×C6 [×4], C2×C6 [×4], C42, C22⋊C4 [×2], C4⋊C4, C22×C4 [×2], C2×D4, C2×Dic3 [×2], C2×Dic3 [×2], C2×Dic3 [×4], C2×C12, C3×D4 [×4], C22×C6 [×2], C4×D4, C4×Dic3, C4⋊Dic3, C6.D4 [×2], C22×Dic3 [×2], C6×D4, D4×Dic3
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×2], C23, Dic3 [×4], D6 [×3], C22×C4, C2×D4, C4○D4, C2×Dic3 [×6], C22×S3, C4×D4, S3×D4, D42S3, C22×Dic3, D4×Dic3

Character table of D4×Dic3

 class 1 2A 2B 2C 2D 2E 2F 2G 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 6A 6B 6C 6D 6E 6F 6G 12A 12B size 1 1 1 1 2 2 2 2 2 2 2 3 3 3 3 6 6 6 6 6 6 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 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 -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 -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 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 -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 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 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 -1 -1 1 1 -1 -1 linear of order 2 ρ9 1 1 -1 -1 1 -1 1 -1 1 1 -1 -i i -i i -i -i i -i i i -1 -1 1 -1 1 1 -1 -1 1 linear of order 4 ρ10 1 1 -1 -1 -1 1 -1 1 1 1 -1 -i i -i i -i i -i i i -i -1 -1 1 1 -1 -1 1 -1 1 linear of order 4 ρ11 1 1 -1 -1 1 -1 1 -1 1 1 -1 i -i i -i i i -i i -i -i -1 -1 1 -1 1 1 -1 -1 1 linear of order 4 ρ12 1 1 -1 -1 -1 1 -1 1 1 1 -1 i -i i -i i -i i -i -i i -1 -1 1 1 -1 -1 1 -1 1 linear of order 4 ρ13 1 1 -1 -1 -1 -1 1 1 1 -1 1 i -i i -i -i i i -i i -i -1 -1 1 -1 1 -1 1 1 -1 linear of order 4 ρ14 1 1 -1 -1 1 1 -1 -1 1 -1 1 i -i i -i -i -i -i i i i -1 -1 1 1 -1 1 -1 1 -1 linear of order 4 ρ15 1 1 -1 -1 -1 -1 1 1 1 -1 1 -i i -i i i -i -i i -i i -1 -1 1 -1 1 -1 1 1 -1 linear of order 4 ρ16 1 1 -1 -1 1 1 -1 -1 1 -1 1 -i i -i i i i i -i -i -i -1 -1 1 1 -1 1 -1 1 -1 linear of order 4 ρ17 2 2 2 2 2 2 2 2 -1 2 2 0 0 0 0 0 0 0 0 0 0 -1 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ18 2 -2 2 -2 0 0 0 0 2 0 0 -2 -2 2 2 0 0 0 0 0 0 -2 2 -2 0 0 0 0 0 0 orthogonal lifted from D4 ρ19 2 2 2 2 2 -2 -2 2 -1 -2 -2 0 0 0 0 0 0 0 0 0 0 -1 -1 -1 1 1 -1 -1 1 1 orthogonal lifted from D6 ρ20 2 2 2 2 -2 2 2 -2 -1 -2 -2 0 0 0 0 0 0 0 0 0 0 -1 -1 -1 -1 -1 1 1 1 1 orthogonal lifted from D6 ρ21 2 -2 2 -2 0 0 0 0 2 0 0 2 2 -2 -2 0 0 0 0 0 0 -2 2 -2 0 0 0 0 0 0 orthogonal lifted from D4 ρ22 2 2 2 2 -2 -2 -2 -2 -1 2 2 0 0 0 0 0 0 0 0 0 0 -1 -1 -1 1 1 1 1 -1 -1 orthogonal lifted from D6 ρ23 2 2 -2 -2 2 2 -2 -2 -1 -2 2 0 0 0 0 0 0 0 0 0 0 1 1 -1 -1 1 -1 1 -1 1 symplectic lifted from Dic3, Schur index 2 ρ24 2 2 -2 -2 2 -2 2 -2 -1 2 -2 0 0 0 0 0 0 0 0 0 0 1 1 -1 1 -1 -1 1 1 -1 symplectic lifted from Dic3, Schur index 2 ρ25 2 2 -2 -2 -2 2 -2 2 -1 2 -2 0 0 0 0 0 0 0 0 0 0 1 1 -1 -1 1 1 -1 1 -1 symplectic lifted from Dic3, Schur index 2 ρ26 2 2 -2 -2 -2 -2 2 2 -1 -2 2 0 0 0 0 0 0 0 0 0 0 1 1 -1 1 -1 1 -1 -1 1 symplectic lifted from Dic3, Schur index 2 ρ27 2 -2 -2 2 0 0 0 0 2 0 0 -2i 2i 2i -2i 0 0 0 0 0 0 2 -2 -2 0 0 0 0 0 0 complex lifted from C4○D4 ρ28 2 -2 -2 2 0 0 0 0 2 0 0 2i -2i -2i 2i 0 0 0 0 0 0 2 -2 -2 0 0 0 0 0 0 complex lifted from C4○D4 ρ29 4 -4 4 -4 0 0 0 0 -2 0 0 0 0 0 0 0 0 0 0 0 0 2 -2 2 0 0 0 0 0 0 orthogonal lifted from S3×D4 ρ30 4 -4 -4 4 0 0 0 0 -2 0 0 0 0 0 0 0 0 0 0 0 0 -2 2 2 0 0 0 0 0 0 symplectic lifted from D4⋊2S3, Schur index 2

Smallest permutation representation of D4×Dic3
On 48 points
Generators in S48
(1 27 16 20)(2 28 17 21)(3 29 18 22)(4 30 13 23)(5 25 14 24)(6 26 15 19)(7 38 43 31)(8 39 44 32)(9 40 45 33)(10 41 46 34)(11 42 47 35)(12 37 48 36)
(1 20)(2 21)(3 22)(4 23)(5 24)(6 19)(7 38)(8 39)(9 40)(10 41)(11 42)(12 37)(13 30)(14 25)(15 26)(16 27)(17 28)(18 29)(31 43)(32 44)(33 45)(34 46)(35 47)(36 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 31 4 34)(2 36 5 33)(3 35 6 32)(7 30 10 27)(8 29 11 26)(9 28 12 25)(13 41 16 38)(14 40 17 37)(15 39 18 42)(19 44 22 47)(20 43 23 46)(21 48 24 45)

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

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

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

Matrix representation of D4×Dic3 in GL5(𝔽13)

 1 0 0 0 0 0 0 1 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 1
,
 12 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 1
,
 12 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 12 1 0 0 0 12 0
,
 5 0 0 0 0 0 5 0 0 0 0 0 5 0 0 0 0 0 1 0 0 0 0 1 12

G:=sub<GL(5,GF(13))| [1,0,0,0,0,0,0,12,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1],[12,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1],[12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12,12,0,0,0,1,0],[5,0,0,0,0,0,5,0,0,0,0,0,5,0,0,0,0,0,1,1,0,0,0,0,12] >;

D4×Dic3 in GAP, Magma, Sage, TeX

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

G:=Group("D4xDic3");
// GroupNames label

G:=SmallGroup(96,141);
// by ID

G=gap.SmallGroup(96,141);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,48,188,2309]);
// Polycyclic

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

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