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## G = Dic3⋊2S4order 288 = 25·32

### The semidirect product of Dic3 and S4 acting through Inn(Dic3)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C3×A4 — Dic3⋊2S4
 Chief series C1 — C22 — C2×C6 — C3×A4 — C6×A4 — Dic3×A4 — Dic3⋊2S4
 Lower central C3×A4 — Dic3⋊2S4
 Upper central C1 — C2

Generators and relations for Dic32S4
G = < a,b,c,d,e,f | a6=c2=d2=e3=f2=1, b2=a3, bab-1=faf=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, bf=fb, ece-1=fcf=cd=dc, ede-1=c, df=fd, fef=e-1 >

Subgroups: 694 in 122 conjugacy classes, 21 normal (19 characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C2×C4, D4, C23, C23, C32, Dic3, Dic3, C12, A4, A4, D6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C3⋊S3, C3×C6, C4×S3, C2×Dic3, C3⋊D4, C2×C12, S4, C2×A4, C2×A4, C22×S3, C22×C6, C4×D4, C3×Dic3, C3×A4, C2×C3⋊S3, C4×Dic3, Dic3⋊C4, D6⋊C4, C3×C22⋊C4, A4⋊C4, C4×A4, S3×C2×C4, C22×Dic3, C2×C3⋊D4, C2×S4, C6.D6, C3⋊S4, C6×A4, Dic34D4, C4×S4, C3×A4⋊C4, Dic3×A4, C2×C3⋊S4, Dic32S4
Quotients: C1, C2, C4, C22, S3, C2×C4, D6, C4×S3, S4, S32, C2×S4, C6.D6, C4×S4, S3×S4, Dic32S4

Character table of Dic32S4

 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 12A 12B 12C 12D 12E 12F size 1 1 3 3 18 18 2 8 16 3 3 6 6 6 6 9 9 18 18 2 6 6 8 16 12 12 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 -i i i -i i -i -1 1 -1 1 -1 -1 -1 i -i -i i i -i linear of order 4 ρ6 1 -1 1 -1 1 -1 1 1 1 -i i -i i i -i -i i 1 -1 -1 1 -1 -1 -1 i -i -i i -i i linear of order 4 ρ7 1 -1 1 -1 -1 1 1 1 1 -i i i -i -i i -i i -1 1 -1 1 -1 -1 -1 -i i i -i -i i linear of order 4 ρ8 1 -1 1 -1 1 -1 1 1 1 i -i i -i -i i i -i 1 -1 -1 1 -1 -1 -1 -i i i -i i -i linear of order 4 ρ9 2 2 2 2 0 0 -1 2 -1 0 0 -2 -2 -2 -2 0 0 0 0 -1 -1 -1 2 -1 1 1 1 1 0 0 orthogonal lifted from D6 ρ10 2 2 2 2 0 0 2 -1 -1 -2 -2 0 0 0 0 -2 -2 0 0 2 2 2 -1 -1 0 0 0 0 1 1 orthogonal lifted from D6 ρ11 2 2 2 2 0 0 -1 2 -1 0 0 2 2 2 2 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 0 0 2 2 0 0 2 2 2 -1 -1 0 0 0 0 -1 -1 orthogonal lifted from S3 ρ13 2 -2 2 -2 0 0 2 -1 -1 -2i 2i 0 0 0 0 -2i 2i 0 0 -2 2 -2 1 1 0 0 0 0 i -i complex lifted from C4×S3 ρ14 2 -2 2 -2 0 0 -1 2 -1 0 0 -2i 2i 2i -2i 0 0 0 0 1 -1 1 -2 1 -i i i -i 0 0 complex lifted from C4×S3 ρ15 2 -2 2 -2 0 0 -1 2 -1 0 0 2i -2i -2i 2i 0 0 0 0 1 -1 1 -2 1 i -i -i i 0 0 complex lifted from C4×S3 ρ16 2 -2 2 -2 0 0 2 -1 -1 2i -2i 0 0 0 0 2i -2i 0 0 -2 2 -2 1 1 0 0 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 0 1 -1 1 -1 0 0 orthogonal lifted from C2×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 0 -1 1 -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 0 1 -1 1 -1 0 0 orthogonal lifted from 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 0 -1 1 -1 1 0 0 orthogonal lifted from S4 ρ21 3 -3 -1 1 -1 1 3 0 0 3i -3i i -i i -i -i i 1 -1 -3 -1 1 0 0 -i -i i i 0 0 complex lifted from C4×S4 ρ22 3 -3 -1 1 1 -1 3 0 0 3i -3i -i i -i i -i i -1 1 -3 -1 1 0 0 i i -i -i 0 0 complex lifted from C4×S4 ρ23 3 -3 -1 1 1 -1 3 0 0 -3i 3i i -i i -i i -i -1 1 -3 -1 1 0 0 -i -i i i 0 0 complex lifted from C4×S4 ρ24 3 -3 -1 1 -1 1 3 0 0 -3i 3i -i i -i i i -i 1 -1 -3 -1 1 0 0 i i -i -i 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 -1 0 0 0 0 0 0 orthogonal lifted from C6.D6 ρ26 4 4 4 4 0 0 -2 -2 1 0 0 0 0 0 0 0 0 0 0 -2 -2 -2 -2 1 0 0 0 0 0 0 orthogonal lifted from S32 ρ27 6 6 -2 -2 0 0 -3 0 0 0 0 2 2 -2 -2 0 0 0 0 -3 1 1 0 0 -1 1 -1 1 0 0 orthogonal lifted from S3×S4 ρ28 6 6 -2 -2 0 0 -3 0 0 0 0 -2 -2 2 2 0 0 0 0 -3 1 1 0 0 1 -1 1 -1 0 0 orthogonal lifted from S3×S4 ρ29 6 -6 -2 2 0 0 -3 0 0 0 0 2i -2i 2i -2i 0 0 0 0 3 1 -1 0 0 i i -i -i 0 0 complex faithful ρ30 6 -6 -2 2 0 0 -3 0 0 0 0 -2i 2i -2i 2i 0 0 0 0 3 1 -1 0 0 -i -i i i 0 0 complex faithful

Smallest permutation representation of Dic32S4
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 20 4 23)(2 19 5 22)(3 24 6 21)(7 36 10 33)(8 35 11 32)(9 34 12 31)(13 30 16 27)(14 29 17 26)(15 28 18 25)
(1 4)(2 5)(3 6)(7 10)(8 11)(9 12)(19 22)(20 23)(21 24)(31 34)(32 35)(33 36)
(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 31 25)(20 32 26)(21 33 27)(22 34 28)(23 35 29)(24 36 30)
(1 4)(2 3)(5 6)(7 18)(8 17)(9 16)(10 15)(11 14)(12 13)(19 24)(20 23)(21 22)(25 36)(26 35)(27 34)(28 33)(29 32)(30 31)

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

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

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

Matrix representation of Dic32S4 in GL5(𝔽13)

 0 12 0 0 0 1 1 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 12
,
 5 0 0 0 0 8 8 0 0 0 0 0 8 0 0 0 0 0 8 0 0 0 0 0 8
,
 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 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 0 0 1 0 0 1 0 0 0 0 0 1 0
,
 1 0 0 0 0 12 12 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 12 0

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

Dic32S4 in GAP, Magma, Sage, TeX

{\rm Dic}_3\rtimes_2S_4
% in TeX

G:=Group("Dic3:2S4");
// GroupNames label

G:=SmallGroup(288,854);
// by ID

G=gap.SmallGroup(288,854);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-3,-2,2,28,36,234,1684,3036,782,1777,1350]);
// 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=f*a*f=a^-1,a*c=c*a,a*d=d*a,a*e=e*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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