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## G = Q8⋊3Dic3order 96 = 25·3

### 2nd semidirect product of Q8 and Dic3 acting via Dic3/C6=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — Q8⋊3Dic3
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C4.Dic3 — Q8⋊3Dic3
 Lower central C3 — C6 — C12 — Q8⋊3Dic3
 Upper central C1 — C4 — C2×C4 — C4○D4

Generators and relations for Q83Dic3
G = < a,b,c,d | a4=c6=1, b2=a2, d2=c3, bab-1=a-1, ac=ca, ad=da, cbc-1=a2b, dbd-1=a-1b, dcd-1=c-1 >

Character table of Q83Dic3

 class 1 2A 2B 2C 3 4A 4B 4C 4D 4E 4F 4G 4H 6A 6B 6C 6D 8A 8B 12A 12B 12C 12D 12E size 1 1 2 4 2 1 1 2 4 6 6 6 6 2 4 4 4 12 12 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 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 -i i 1 -1 -1 -1 -i i -1 -1 1 1 1 linear of order 4 ρ6 1 1 -1 -1 1 -1 -1 1 1 i -i i -i 1 -1 -1 -1 i -i -1 -1 1 1 1 linear of order 4 ρ7 1 1 -1 1 1 -1 -1 1 -1 i -i i -i 1 1 -1 1 -i i -1 -1 -1 1 -1 linear of order 4 ρ8 1 1 -1 1 1 -1 -1 1 -1 -i i -i i 1 1 -1 1 i -i -1 -1 -1 1 -1 linear of order 4 ρ9 2 2 -2 0 2 2 2 -2 0 0 0 0 0 2 0 -2 0 0 0 2 2 0 -2 0 orthogonal lifted from D4 ρ10 2 2 2 -2 -1 2 2 2 -2 0 0 0 0 -1 1 -1 1 0 0 -1 -1 1 -1 1 orthogonal lifted from D6 ρ11 2 2 2 0 2 -2 -2 -2 0 0 0 0 0 2 0 2 0 0 0 -2 -2 0 -2 0 orthogonal lifted from D4 ρ12 2 2 2 2 -1 2 2 2 2 0 0 0 0 -1 -1 -1 -1 0 0 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ13 2 2 -2 -2 -1 -2 -2 2 2 0 0 0 0 -1 1 1 1 0 0 1 1 -1 -1 -1 symplectic lifted from Dic3, Schur index 2 ρ14 2 2 -2 2 -1 -2 -2 2 -2 0 0 0 0 -1 -1 1 -1 0 0 1 1 1 -1 1 symplectic lifted from Dic3, Schur index 2 ρ15 2 2 -2 0 -1 2 2 -2 0 0 0 0 0 -1 √-3 1 -√-3 0 0 -1 -1 -√-3 1 √-3 complex lifted from C3⋊D4 ρ16 2 2 2 0 -1 -2 -2 -2 0 0 0 0 0 -1 √-3 -1 -√-3 0 0 1 1 √-3 1 -√-3 complex lifted from C3⋊D4 ρ17 2 2 2 0 -1 -2 -2 -2 0 0 0 0 0 -1 -√-3 -1 √-3 0 0 1 1 -√-3 1 √-3 complex lifted from C3⋊D4 ρ18 2 2 -2 0 -1 2 2 -2 0 0 0 0 0 -1 -√-3 1 √-3 0 0 -1 -1 √-3 1 -√-3 complex lifted from C3⋊D4 ρ19 2 -2 0 0 2 -2i 2i 0 0 1-i -1-i -1+i 1+i -2 0 0 0 0 0 2i -2i 0 0 0 complex lifted from C4≀C2 ρ20 2 -2 0 0 2 -2i 2i 0 0 -1+i 1+i 1-i -1-i -2 0 0 0 0 0 2i -2i 0 0 0 complex lifted from C4≀C2 ρ21 2 -2 0 0 2 2i -2i 0 0 -1-i 1-i 1+i -1+i -2 0 0 0 0 0 -2i 2i 0 0 0 complex lifted from C4≀C2 ρ22 2 -2 0 0 2 2i -2i 0 0 1+i -1+i -1-i 1-i -2 0 0 0 0 0 -2i 2i 0 0 0 complex lifted from C4≀C2 ρ23 4 -4 0 0 -2 4i -4i 0 0 0 0 0 0 2 0 0 0 0 0 2i -2i 0 0 0 complex faithful ρ24 4 -4 0 0 -2 -4i 4i 0 0 0 0 0 0 2 0 0 0 0 0 -2i 2i 0 0 0 complex faithful

Permutation representations of Q83Dic3
On 24 points - transitive group 24T109
Generators in S24
(1 10 6 7)(2 11 4 8)(3 12 5 9)(13 22 16 19)(14 23 17 20)(15 24 18 21)
(1 22 6 19)(2 20 4 23)(3 24 5 21)(7 16 10 13)(8 14 11 17)(9 18 12 15)
(1 2 3)(4 5 6)(7 8 9)(10 11 12)(13 14 15 16 17 18)(19 20 21 22 23 24)
(2 3)(4 5)(8 9)(11 12)(13 19 16 22)(14 24 17 21)(15 23 18 20)

G:=sub<Sym(24)| (1,10,6,7)(2,11,4,8)(3,12,5,9)(13,22,16,19)(14,23,17,20)(15,24,18,21), (1,22,6,19)(2,20,4,23)(3,24,5,21)(7,16,10,13)(8,14,11,17)(9,18,12,15), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24), (2,3)(4,5)(8,9)(11,12)(13,19,16,22)(14,24,17,21)(15,23,18,20)>;

G:=Group( (1,10,6,7)(2,11,4,8)(3,12,5,9)(13,22,16,19)(14,23,17,20)(15,24,18,21), (1,22,6,19)(2,20,4,23)(3,24,5,21)(7,16,10,13)(8,14,11,17)(9,18,12,15), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24), (2,3)(4,5)(8,9)(11,12)(13,19,16,22)(14,24,17,21)(15,23,18,20) );

G=PermutationGroup([[(1,10,6,7),(2,11,4,8),(3,12,5,9),(13,22,16,19),(14,23,17,20),(15,24,18,21)], [(1,22,6,19),(2,20,4,23),(3,24,5,21),(7,16,10,13),(8,14,11,17),(9,18,12,15)], [(1,2,3),(4,5,6),(7,8,9),(10,11,12),(13,14,15,16,17,18),(19,20,21,22,23,24)], [(2,3),(4,5),(8,9),(11,12),(13,19,16,22),(14,24,17,21),(15,23,18,20)]])

G:=TransitiveGroup(24,109);

Matrix representation of Q83Dic3 in GL4(𝔽5) generated by

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

Q83Dic3 in GAP, Magma, Sage, TeX

Q_8\rtimes_3{\rm Dic}_3
% in TeX

G:=Group("Q8:3Dic3");
// GroupNames label

G:=SmallGroup(96,44);
// by ID

G=gap.SmallGroup(96,44);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,24,121,86,579,297,69,2309]);
// Polycyclic

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

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