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## G = Dic3.D6order 144 = 24·32

### 2nd non-split extension by Dic3 of D6 acting via D6/S3=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — Dic3.D6
 Chief series C1 — C3 — C32 — C3×C6 — C3×Dic3 — C6.D6 — Dic3.D6
 Lower central C32 — C3×C6 — Dic3.D6
 Upper central C1 — C2 — C4

Generators and relations for Dic3.D6
G = < a,b,c,d | a6=1, b2=c6=d2=a3, bab-1=a-1, ac=ca, ad=da, cbc-1=dbd-1=a3b, dcd-1=a3c5 >

Subgroups: 256 in 84 conjugacy classes, 34 normal (10 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C2×C4, Q8, C32, Dic3, Dic3, C12, C12, D6, C2×Q8, C3⋊S3, C3×C6, Dic6, Dic6, C4×S3, C3×Q8, C3×Dic3, C3⋊Dic3, C3×C12, C2×C3⋊S3, S3×Q8, C6.D6, C322Q8, C3×Dic6, C4×C3⋊S3, Dic3.D6
Quotients: C1, C2, C22, S3, Q8, C23, D6, C2×Q8, C22×S3, S32, S3×Q8, C2×S32, Dic3.D6

Character table of Dic3.D6

 class 1 2A 2B 2C 3A 3B 3C 4A 4B 4C 4D 4E 4F 6A 6B 6C 12A 12B 12C 12D 12E 12F 12G 12H size 1 1 9 9 2 2 4 2 6 6 6 6 18 2 2 4 4 4 4 4 12 12 12 12 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 0 0 2 -1 -1 -2 -2 2 0 0 0 -1 2 -1 1 1 1 -2 -1 0 1 0 orthogonal lifted from D6 ρ10 2 2 0 0 -1 2 -1 2 0 0 2 2 0 2 -1 -1 -1 2 -1 -1 0 -1 0 -1 orthogonal lifted from S3 ρ11 2 2 0 0 2 -1 -1 -2 2 -2 0 0 0 -1 2 -1 1 1 1 -2 1 0 -1 0 orthogonal lifted from D6 ρ12 2 2 0 0 2 -1 -1 2 -2 -2 0 0 0 -1 2 -1 -1 -1 -1 2 1 0 1 0 orthogonal lifted from D6 ρ13 2 2 0 0 -1 2 -1 -2 0 0 -2 2 0 2 -1 -1 1 -2 1 1 0 -1 0 1 orthogonal lifted from D6 ρ14 2 2 0 0 -1 2 -1 2 0 0 -2 -2 0 2 -1 -1 -1 2 -1 -1 0 1 0 1 orthogonal lifted from D6 ρ15 2 2 0 0 -1 2 -1 -2 0 0 2 -2 0 2 -1 -1 1 -2 1 1 0 1 0 -1 orthogonal lifted from D6 ρ16 2 2 0 0 2 -1 -1 2 2 2 0 0 0 -1 2 -1 -1 -1 -1 2 -1 0 -1 0 orthogonal lifted from S3 ρ17 2 -2 2 -2 2 2 2 0 0 0 0 0 0 -2 -2 -2 0 0 0 0 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ18 2 -2 -2 2 2 2 2 0 0 0 0 0 0 -2 -2 -2 0 0 0 0 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ19 4 4 0 0 -2 -2 1 -4 0 0 0 0 0 -2 -2 1 -1 2 -1 2 0 0 0 0 orthogonal lifted from C2×S32 ρ20 4 4 0 0 -2 -2 1 4 0 0 0 0 0 -2 -2 1 1 -2 1 -2 0 0 0 0 orthogonal lifted from S32 ρ21 4 -4 0 0 -2 4 -2 0 0 0 0 0 0 -4 2 2 0 0 0 0 0 0 0 0 symplectic lifted from S3×Q8, Schur index 2 ρ22 4 -4 0 0 4 -2 -2 0 0 0 0 0 0 2 -4 2 0 0 0 0 0 0 0 0 symplectic lifted from S3×Q8, Schur index 2 ρ23 4 -4 0 0 -2 -2 1 0 0 0 0 0 0 2 2 -1 3i 0 -3i 0 0 0 0 0 complex faithful ρ24 4 -4 0 0 -2 -2 1 0 0 0 0 0 0 2 2 -1 -3i 0 3i 0 0 0 0 0 complex faithful

Permutation representations of Dic3.D6
On 24 points - transitive group 24T223
Generators in S24
(1 11 9 7 5 3)(2 12 10 8 6 4)(13 15 17 19 21 23)(14 16 18 20 22 24)
(1 22 7 16)(2 17 8 23)(3 24 9 18)(4 19 10 13)(5 14 11 20)(6 21 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)
(1 21 7 15)(2 20 8 14)(3 19 9 13)(4 18 10 24)(5 17 11 23)(6 16 12 22)

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

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

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

G:=TransitiveGroup(24,223);

Matrix representation of Dic3.D6 in GL4(𝔽5) generated by

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

Dic3.D6 in GAP, Magma, Sage, TeX

{\rm Dic}_3.D_6
% in TeX

G:=Group("Dic3.D6");
// GroupNames label

G:=SmallGroup(144,140);
// by ID

G=gap.SmallGroup(144,140);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-3,48,55,218,116,50,490,3461]);
// Polycyclic

G:=Group<a,b,c,d|a^6=1,b^2=c^6=d^2=a^3,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=a^3*b,d*c*d^-1=a^3*c^5>;
// generators/relations

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