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## G = D4.Dic3order 96 = 25·3

### The non-split extension by D4 of Dic3 acting through Inn(D4)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — D4.Dic3
 Chief series C1 — C3 — C6 — C12 — C3⋊C8 — C2×C3⋊C8 — D4.Dic3
 Lower central C3 — C6 — D4.Dic3
 Upper central C1 — C4 — C4○D4

Generators and relations for D4.Dic3
G = < a,b,c,d | a4=b2=1, c6=a2, d2=a2c3, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c5 >

Subgroups: 90 in 62 conjugacy classes, 45 normal (12 characteristic)
C1, C2, C2, C3, C4, C4, C22, C6, C6, C8, C2×C4, D4, Q8, C12, C12, C2×C6, C2×C8, M4(2), C4○D4, C3⋊C8, C3⋊C8, C2×C12, C3×D4, C3×Q8, C8○D4, C2×C3⋊C8, C4.Dic3, C3×C4○D4, D4.Dic3
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, Dic3, D6, C22×C4, C2×Dic3, C22×S3, C8○D4, C22×Dic3, D4.Dic3

Character table of D4.Dic3

 class 1 2A 2B 2C 2D 3 4A 4B 4C 4D 4E 6A 6B 6C 6D 8A 8B 8C 8D 8E 8F 8G 8H 8I 8J 12A 12B 12C 12D 12E size 1 1 2 2 2 2 1 1 2 2 2 2 4 4 4 3 3 3 3 6 6 6 6 6 6 2 2 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 1 1 1 -1 -i i -i i -i -i i -i i i -1 -1 1 -1 -1 linear of order 4 ρ10 1 1 -1 1 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 linear of order 4 ρ11 1 1 -1 -1 -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 linear of order 4 ρ12 1 1 1 -1 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 linear of order 4 ρ13 1 1 -1 1 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 linear of order 4 ρ14 1 1 1 1 -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 linear of order 4 ρ15 1 1 1 -1 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 linear of order 4 ρ16 1 1 -1 -1 -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 linear of order 4 ρ17 2 2 -2 2 -2 -1 2 2 -2 2 -2 -1 -1 1 1 0 0 0 0 0 0 0 0 0 0 -1 -1 1 1 -1 orthogonal lifted from D6 ρ18 2 2 2 2 2 -1 2 2 2 2 2 -1 -1 -1 -1 0 0 0 0 0 0 0 0 0 0 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ19 2 2 2 -2 -2 -1 2 2 -2 -2 2 -1 1 -1 1 0 0 0 0 0 0 0 0 0 0 -1 -1 1 -1 1 orthogonal lifted from D6 ρ20 2 2 -2 -2 2 -1 2 2 2 -2 -2 -1 1 1 -1 0 0 0 0 0 0 0 0 0 0 -1 -1 -1 1 1 orthogonal lifted from D6 ρ21 2 2 2 -2 2 -1 -2 -2 -2 2 -2 -1 1 -1 -1 0 0 0 0 0 0 0 0 0 0 1 1 1 1 -1 symplectic lifted from Dic3, Schur index 2 ρ22 2 2 -2 -2 -2 -1 -2 -2 2 2 2 -1 1 1 1 0 0 0 0 0 0 0 0 0 0 1 1 -1 -1 -1 symplectic lifted from Dic3, Schur index 2 ρ23 2 2 -2 2 2 -1 -2 -2 -2 -2 2 -1 -1 1 -1 0 0 0 0 0 0 0 0 0 0 1 1 1 -1 1 symplectic lifted from Dic3, Schur index 2 ρ24 2 2 2 2 -2 -1 -2 -2 2 -2 -2 -1 -1 -1 1 0 0 0 0 0 0 0 0 0 0 1 1 -1 1 1 symplectic lifted from Dic3, Schur index 2 ρ25 2 -2 0 0 0 2 2i -2i 0 0 0 -2 0 0 0 2ζ83 2ζ8 2ζ87 2ζ85 0 0 0 0 0 0 -2i 2i 0 0 0 complex lifted from C8○D4 ρ26 2 -2 0 0 0 2 -2i 2i 0 0 0 -2 0 0 0 2ζ85 2ζ87 2ζ8 2ζ83 0 0 0 0 0 0 2i -2i 0 0 0 complex lifted from C8○D4 ρ27 2 -2 0 0 0 2 -2i 2i 0 0 0 -2 0 0 0 2ζ8 2ζ83 2ζ85 2ζ87 0 0 0 0 0 0 2i -2i 0 0 0 complex lifted from C8○D4 ρ28 2 -2 0 0 0 2 2i -2i 0 0 0 -2 0 0 0 2ζ87 2ζ85 2ζ83 2ζ8 0 0 0 0 0 0 -2i 2i 0 0 0 complex lifted from C8○D4 ρ29 4 -4 0 0 0 -2 -4i 4i 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 -2i 2i 0 0 0 complex faithful ρ30 4 -4 0 0 0 -2 4i -4i 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 2i -2i 0 0 0 complex faithful

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

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

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

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

Matrix representation of D4.Dic3 in GL4(𝔽5) generated by

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

D4.Dic3 in GAP, Magma, Sage, TeX

D_4.{\rm Dic}_3
% in TeX

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

G:=SmallGroup(96,155);
// by ID

G=gap.SmallGroup(96,155);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,48,188,69,2309]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^2=1,c^6=a^2,d^2=a^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^5>;
// generators/relations

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