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G = Dic3×S4order 288 = 25·32

Direct product of Dic3 and S4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C3×A4 — Dic3×S4
 Chief series C1 — C22 — C2×C6 — C3×A4 — C6×A4 — Dic3×A4 — Dic3×S4
 Lower central C3×A4 — Dic3×S4
 Upper central C1 — C2

Generators and relations for Dic3×S4
G = < a,b,c,d,e,f | a6=c2=d2=e3=f2=1, b2=a3, bab-1=a-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, ece-1=fcf=cd=dc, ede-1=c, df=fd, fef=e-1 >

Subgroups: 542 in 118 conjugacy classes, 23 normal (19 characteristic)
C1, C2, C2 [×4], C3, C3 [×2], C4 [×6], C22, C22 [×6], S3 [×2], C6, C6 [×6], C2×C4 [×7], D4 [×4], C23, C23, C32, Dic3, Dic3 [×5], C12 [×3], A4, A4, D6, C2×C6, C2×C6 [×6], C42, C22⋊C4 [×2], C4⋊C4, C22×C4 [×2], C2×D4, C3×S3 [×2], C3×C6, C4×S3, C2×Dic3 [×6], C2×C12, C3×D4 [×4], S4 [×2], C2×A4, C2×A4, C22×C6, C22×C6, C4×D4, C3×Dic3, C3⋊Dic3, C3×A4, S3×C6, C4×Dic3, C4⋊Dic3, C6.D4 [×2], A4⋊C4 [×2], C4×A4, C22×Dic3, C22×Dic3, C6×D4, C2×S4, S3×Dic3, C3×S4 [×2], C6×A4, D4×Dic3, C4×S4, C6.7S4, Dic3×A4, C6×S4, Dic3×S4
Quotients: C1, C2 [×3], C4 [×2], C22, S3 [×2], C2×C4, Dic3 [×2], D6 [×2], C4×S3, C2×Dic3, S4, S32, C2×S4, S3×Dic3, C4×S4, S3×S4, Dic3×S4

Character table of Dic3×S4

 class 1 2A 2B 2C 2D 2E 3A 3B 3C 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 6A 6B 6C 6D 6E 6F 6G 12A 12B 12C 12D size 1 1 3 3 6 6 2 8 16 3 3 6 6 9 9 18 18 18 18 2 6 6 8 12 12 16 12 12 24 24 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 -i i -1 1 i -i -i i -i i -1 1 -1 -1 -1 1 -1 1 -1 i -i linear of order 4 ρ6 1 -1 1 -1 -1 1 1 1 1 -i i 1 -1 i -i i -i i -i -1 1 -1 -1 1 -1 -1 -1 1 i -i linear of order 4 ρ7 1 -1 1 -1 1 -1 1 1 1 i -i -1 1 -i i i -i i -i -1 1 -1 -1 -1 1 -1 1 -1 -i i linear of order 4 ρ8 1 -1 1 -1 -1 1 1 1 1 i -i 1 -1 -i i -i i -i i -1 1 -1 -1 1 -1 -1 -1 1 -i i linear of order 4 ρ9 2 2 2 2 0 0 2 -1 -1 -2 -2 0 0 -2 -2 0 0 0 0 2 2 2 -1 0 0 -1 0 0 1 1 orthogonal lifted from D6 ρ10 2 2 2 2 -2 -2 -1 2 -1 0 0 -2 -2 0 0 0 0 0 0 -1 -1 -1 2 1 1 -1 1 1 0 0 orthogonal lifted from D6 ρ11 2 2 2 2 2 2 -1 2 -1 0 0 2 2 0 0 0 0 0 0 -1 -1 -1 2 -1 -1 -1 -1 -1 0 0 orthogonal lifted from S3 ρ12 2 2 2 2 0 0 2 -1 -1 2 2 0 0 2 2 0 0 0 0 2 2 2 -1 0 0 -1 0 0 -1 -1 orthogonal lifted from S3 ρ13 2 -2 2 -2 2 -2 -1 2 -1 0 0 -2 2 0 0 0 0 0 0 1 -1 1 -2 1 -1 1 -1 1 0 0 symplectic lifted from Dic3, Schur index 2 ρ14 2 -2 2 -2 -2 2 -1 2 -1 0 0 2 -2 0 0 0 0 0 0 1 -1 1 -2 -1 1 1 1 -1 0 0 symplectic lifted from Dic3, Schur index 2 ρ15 2 -2 2 -2 0 0 2 -1 -1 2i -2i 0 0 -2i 2i 0 0 0 0 -2 2 -2 1 0 0 1 0 0 i -i complex lifted from C4×S3 ρ16 2 -2 2 -2 0 0 2 -1 -1 -2i 2i 0 0 2i -2i 0 0 0 0 -2 2 -2 1 0 0 1 0 0 -i i complex lifted from C4×S3 ρ17 3 3 -1 -1 1 1 3 0 0 3 3 -1 -1 -1 -1 -1 -1 1 1 3 -1 -1 0 1 1 0 -1 -1 0 0 orthogonal lifted from S4 ρ18 3 3 -1 -1 1 1 3 0 0 -3 -3 -1 -1 1 1 1 1 -1 -1 3 -1 -1 0 1 1 0 -1 -1 0 0 orthogonal lifted from C2×S4 ρ19 3 3 -1 -1 -1 -1 3 0 0 -3 -3 1 1 1 1 -1 -1 1 1 3 -1 -1 0 -1 -1 0 1 1 0 0 orthogonal lifted from C2×S4 ρ20 3 3 -1 -1 -1 -1 3 0 0 3 3 1 1 -1 -1 1 1 -1 -1 3 -1 -1 0 -1 -1 0 1 1 0 0 orthogonal lifted from S4 ρ21 3 -3 -1 1 1 -1 3 0 0 3i -3i 1 -1 i -i -i i i -i -3 -1 1 0 -1 1 0 -1 1 0 0 complex lifted from C4×S4 ρ22 3 -3 -1 1 1 -1 3 0 0 -3i 3i 1 -1 -i i i -i -i i -3 -1 1 0 -1 1 0 -1 1 0 0 complex lifted from C4×S4 ρ23 3 -3 -1 1 -1 1 3 0 0 -3i 3i -1 1 -i i -i i i -i -3 -1 1 0 1 -1 0 1 -1 0 0 complex lifted from C4×S4 ρ24 3 -3 -1 1 -1 1 3 0 0 3i -3i -1 1 i -i i -i -i i -3 -1 1 0 1 -1 0 1 -1 0 0 complex lifted from C4×S4 ρ25 4 4 4 4 0 0 -2 -2 1 0 0 0 0 0 0 0 0 0 0 -2 -2 -2 -2 0 0 1 0 0 0 0 orthogonal lifted from S32 ρ26 4 -4 4 -4 0 0 -2 -2 1 0 0 0 0 0 0 0 0 0 0 2 -2 2 2 0 0 -1 0 0 0 0 symplectic lifted from S3×Dic3, Schur index 2 ρ27 6 6 -2 -2 2 2 -3 0 0 0 0 -2 -2 0 0 0 0 0 0 -3 1 1 0 -1 -1 0 1 1 0 0 orthogonal lifted from S3×S4 ρ28 6 6 -2 -2 -2 -2 -3 0 0 0 0 2 2 0 0 0 0 0 0 -3 1 1 0 1 1 0 -1 -1 0 0 orthogonal lifted from S3×S4 ρ29 6 -6 -2 2 2 -2 -3 0 0 0 0 2 -2 0 0 0 0 0 0 3 1 -1 0 1 -1 0 1 -1 0 0 symplectic faithful, Schur index 2 ρ30 6 -6 -2 2 -2 2 -3 0 0 0 0 -2 2 0 0 0 0 0 0 3 1 -1 0 -1 1 0 -1 1 0 0 symplectic faithful, Schur index 2

Smallest permutation representation of Dic3×S4
On 36 points
Generators in S36
(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)
(1 22 4 19)(2 21 5 24)(3 20 6 23)(7 26 10 29)(8 25 11 28)(9 30 12 27)(13 32 16 35)(14 31 17 34)(15 36 18 33)
(1 4)(2 5)(3 6)(7 10)(8 11)(9 12)(19 22)(20 23)(21 24)(25 28)(26 29)(27 30)
(7 10)(8 11)(9 12)(13 16)(14 17)(15 18)(25 28)(26 29)(27 30)(31 34)(32 35)(33 36)
(1 11 17)(2 12 18)(3 7 13)(4 8 14)(5 9 15)(6 10 16)(19 25 31)(20 26 32)(21 27 33)(22 28 34)(23 29 35)(24 30 36)
(7 13)(8 14)(9 15)(10 16)(11 17)(12 18)(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)

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

G:=Group( (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), (1,22,4,19)(2,21,5,24)(3,20,6,23)(7,26,10,29)(8,25,11,28)(9,30,12,27)(13,32,16,35)(14,31,17,34)(15,36,18,33), (1,4)(2,5)(3,6)(7,10)(8,11)(9,12)(19,22)(20,23)(21,24)(25,28)(26,29)(27,30), (7,10)(8,11)(9,12)(13,16)(14,17)(15,18)(25,28)(26,29)(27,30)(31,34)(32,35)(33,36), (1,11,17)(2,12,18)(3,7,13)(4,8,14)(5,9,15)(6,10,16)(19,25,31)(20,26,32)(21,27,33)(22,28,34)(23,29,35)(24,30,36), (7,13)(8,14)(9,15)(10,16)(11,17)(12,18)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36) );

G=PermutationGroup([(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)], [(1,22,4,19),(2,21,5,24),(3,20,6,23),(7,26,10,29),(8,25,11,28),(9,30,12,27),(13,32,16,35),(14,31,17,34),(15,36,18,33)], [(1,4),(2,5),(3,6),(7,10),(8,11),(9,12),(19,22),(20,23),(21,24),(25,28),(26,29),(27,30)], [(7,10),(8,11),(9,12),(13,16),(14,17),(15,18),(25,28),(26,29),(27,30),(31,34),(32,35),(33,36)], [(1,11,17),(2,12,18),(3,7,13),(4,8,14),(5,9,15),(6,10,16),(19,25,31),(20,26,32),(21,27,33),(22,28,34),(23,29,35),(24,30,36)], [(7,13),(8,14),(9,15),(10,16),(11,17),(12,18),(25,31),(26,32),(27,33),(28,34),(29,35),(30,36)])

Matrix representation of Dic3×S4 in GL5(𝔽13)

 4 0 0 0 0 9 10 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 5 12 0 0 0 0 8 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 12
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 12 0 0 0 0 0 12
,
 1 0 0 0 0 0 1 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 12 0 0 1 0 0
,
 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 12

G:=sub<GL(5,GF(13))| [4,9,0,0,0,0,10,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[5,0,0,0,0,12,8,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,12,0,0,0,0,0,12,0],[1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,12] >;

Dic3×S4 in GAP, Magma, Sage, TeX

{\rm Dic}_3\times S_4
% in TeX

G:=Group("Dic3xS4");
// GroupNames label

G:=SmallGroup(288,853);
// by ID

G=gap.SmallGroup(288,853);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-3,-2,2,28,234,1684,3036,782,1777,1350]);
// Polycyclic

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

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