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## G = D4⋊Dic3order 96 = 25·3

### 1st semidirect product of D4 and Dic3 acting via Dic3/C6=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — D4⋊Dic3
 Chief series C1 — C3 — C6 — C2×C6 — C2×C12 — C4⋊Dic3 — D4⋊Dic3
 Lower central C3 — C6 — C12 — D4⋊Dic3
 Upper central C1 — C22 — C2×C4 — C2×D4

Generators and relations for D4⋊Dic3
G = < a,b,c,d | a4=b2=c6=1, d2=c3, bab=dad-1=a-1, ac=ca, bc=cb, dbd-1=ab, dcd-1=c-1 >

Character table of D4⋊Dic3

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 6A 6B 6C 6D 6E 6F 6G 8A 8B 8C 8D 12A 12B size 1 1 1 1 4 4 2 2 2 12 12 2 2 2 4 4 4 4 6 6 6 6 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 i -i 1 -1 -1 1 1 -1 -1 -i i -i i -1 1 linear of order 4 ρ6 1 1 -1 -1 -1 1 1 -1 1 -i i 1 -1 -1 1 1 -1 -1 i -i i -i -1 1 linear of order 4 ρ7 1 1 -1 -1 1 -1 1 -1 1 i -i 1 -1 -1 -1 -1 1 1 i -i i -i -1 1 linear of order 4 ρ8 1 1 -1 -1 1 -1 1 -1 1 -i i 1 -1 -1 -1 -1 1 1 -i i -i i -1 1 linear of order 4 ρ9 2 2 2 2 0 0 2 -2 -2 0 0 2 2 2 0 0 0 0 0 0 0 0 -2 -2 orthogonal lifted from D4 ρ10 2 2 2 2 2 2 -1 2 2 0 0 -1 -1 -1 -1 -1 -1 -1 0 0 0 0 -1 -1 orthogonal lifted from S3 ρ11 2 -2 2 -2 0 0 2 0 0 0 0 -2 -2 2 0 0 0 0 √2 √2 -√2 -√2 0 0 orthogonal lifted from D8 ρ12 2 2 2 2 -2 -2 -1 2 2 0 0 -1 -1 -1 1 1 1 1 0 0 0 0 -1 -1 orthogonal lifted from D6 ρ13 2 2 -2 -2 0 0 2 2 -2 0 0 2 -2 -2 0 0 0 0 0 0 0 0 2 -2 orthogonal lifted from D4 ρ14 2 -2 2 -2 0 0 2 0 0 0 0 -2 -2 2 0 0 0 0 -√2 -√2 √2 √2 0 0 orthogonal lifted from D8 ρ15 2 2 -2 -2 2 -2 -1 -2 2 0 0 -1 1 1 1 1 -1 -1 0 0 0 0 1 -1 symplectic lifted from Dic3, Schur index 2 ρ16 2 2 -2 -2 -2 2 -1 -2 2 0 0 -1 1 1 -1 -1 1 1 0 0 0 0 1 -1 symplectic lifted from Dic3, Schur index 2 ρ17 2 2 -2 -2 0 0 -1 2 -2 0 0 -1 1 1 √-3 -√-3 -√-3 √-3 0 0 0 0 -1 1 complex lifted from C3⋊D4 ρ18 2 2 2 2 0 0 -1 -2 -2 0 0 -1 -1 -1 -√-3 √-3 -√-3 √-3 0 0 0 0 1 1 complex lifted from C3⋊D4 ρ19 2 2 -2 -2 0 0 -1 2 -2 0 0 -1 1 1 -√-3 √-3 √-3 -√-3 0 0 0 0 -1 1 complex lifted from C3⋊D4 ρ20 2 -2 -2 2 0 0 2 0 0 0 0 -2 2 -2 0 0 0 0 -√-2 √-2 √-2 -√-2 0 0 complex lifted from SD16 ρ21 2 -2 -2 2 0 0 2 0 0 0 0 -2 2 -2 0 0 0 0 √-2 -√-2 -√-2 √-2 0 0 complex lifted from SD16 ρ22 2 2 2 2 0 0 -1 -2 -2 0 0 -1 -1 -1 √-3 -√-3 √-3 -√-3 0 0 0 0 1 1 complex lifted from C3⋊D4 ρ23 4 -4 4 -4 0 0 -2 0 0 0 0 2 2 -2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4⋊S3, Schur index 2 ρ24 4 -4 -4 4 0 0 -2 0 0 0 0 2 -2 2 0 0 0 0 0 0 0 0 0 0 symplectic lifted from D4.S3, Schur index 2

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

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

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

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

Matrix representation of D4⋊Dic3 in GL4(𝔽73) generated by

 1 0 0 0 0 1 0 0 0 0 1 66 0 0 42 72
,
 1 0 0 0 0 1 0 0 0 0 1 66 0 0 0 72
,
 65 0 0 0 71 9 0 0 0 0 1 0 0 0 0 1
,
 27 26 0 0 0 46 0 0 0 0 0 39 0 0 15 0
G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,1,42,0,0,66,72],[1,0,0,0,0,1,0,0,0,0,1,0,0,0,66,72],[65,71,0,0,0,9,0,0,0,0,1,0,0,0,0,1],[27,0,0,0,26,46,0,0,0,0,0,15,0,0,39,0] >;

D4⋊Dic3 in GAP, Magma, Sage, TeX

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

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

G:=SmallGroup(96,39);
// by ID

G=gap.SmallGroup(96,39);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,24,121,579,297,69,2309]);
// Polycyclic

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

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