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## G = S3×A4⋊C4order 288 = 25·32

### Direct product of S3 and A4⋊C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C3×A4 — S3×A4⋊C4
 Chief series C1 — C22 — C2×C6 — C3×A4 — C6×A4 — C2×S3×A4 — S3×A4⋊C4
 Lower central C3×A4 — S3×A4⋊C4
 Upper central C1 — C2

Generators and relations for S3×A4⋊C4
G = < a,b,c,d,e,f | a3=b2=c2=d2=e3=f4=1, bab=a-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, ece-1=fcf-1=cd=dc, ede-1=c, df=fd, fef-1=e-1 >

Subgroups: 698 in 132 conjugacy classes, 25 normal (19 characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, S3, S3, C6, C6, C2×C4, C23, C23, C32, Dic3, C12, A4, A4, D6, D6, C2×C6, C2×C6, C22⋊C4, C22×C4, C24, C3×S3, C3×C6, C4×S3, C2×Dic3, C2×C12, C2×A4, C2×A4, C22×S3, C22×S3, C22×C6, C2×C22⋊C4, C3×Dic3, C3⋊Dic3, C3×A4, S3×C6, D6⋊C4, C6.D4, C3×C22⋊C4, A4⋊C4, A4⋊C4, S3×C2×C4, C22×A4, S3×C23, S3×Dic3, S3×A4, C6×A4, S3×C22⋊C4, C2×A4⋊C4, C3×A4⋊C4, C6.7S4, C2×S3×A4, S3×A4⋊C4
Quotients: C1, C2, C4, C22, S3, C2×C4, Dic3, D6, C4×S3, C2×Dic3, S4, S32, A4⋊C4, C2×S4, S3×Dic3, C2×A4⋊C4, S3×S4, S3×A4⋊C4

Character table of S3×A4⋊C4

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 4A 4B 4C 4D 4E 4F 4G 4H 6A 6B 6C 6D 6E 6F 6G 12A 12B 12C 12D size 1 1 3 3 3 3 9 9 2 8 16 6 6 6 6 18 18 18 18 2 6 6 8 16 24 24 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 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 i -i -i i -i i -i i -1 1 -1 -1 -1 -1 1 -i i i -i linear of order 4 ρ6 1 -1 -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 linear of order 4 ρ7 1 -1 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 linear of order 4 ρ8 1 -1 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 linear of order 4 ρ9 2 2 0 2 0 2 0 0 -1 2 -1 -2 -2 -2 -2 0 0 0 0 -1 -1 -1 2 -1 0 0 1 1 1 1 orthogonal lifted from D6 ρ10 2 2 -2 2 -2 2 -2 -2 2 -1 -1 0 0 0 0 0 0 0 0 2 2 2 -1 -1 1 1 0 0 0 0 orthogonal lifted from D6 ρ11 2 2 0 2 0 2 0 0 -1 2 -1 2 2 2 2 0 0 0 0 -1 -1 -1 2 -1 0 0 -1 -1 -1 -1 orthogonal lifted from S3 ρ12 2 2 2 2 2 2 2 2 2 -1 -1 0 0 0 0 0 0 0 0 2 2 2 -1 -1 -1 -1 0 0 0 0 orthogonal lifted from S3 ρ13 2 -2 2 -2 -2 2 2 -2 2 -1 -1 0 0 0 0 0 0 0 0 -2 2 -2 1 1 -1 1 0 0 0 0 symplectic lifted from Dic3, Schur index 2 ρ14 2 -2 -2 -2 2 2 -2 2 2 -1 -1 0 0 0 0 0 0 0 0 -2 2 -2 1 1 1 -1 0 0 0 0 symplectic lifted from Dic3, Schur index 2 ρ15 2 -2 0 -2 0 2 0 0 -1 2 -1 2i -2i -2i 2i 0 0 0 0 1 -1 1 -2 1 0 0 i -i -i i complex lifted from C4×S3 ρ16 2 -2 0 -2 0 2 0 0 -1 2 -1 -2i 2i 2i -2i 0 0 0 0 1 -1 1 -2 1 0 0 -i i i -i complex lifted from C4×S3 ρ17 3 3 3 -1 3 -1 -1 -1 3 0 0 1 1 -1 -1 1 1 -1 -1 3 -1 -1 0 0 0 0 1 -1 1 -1 orthogonal lifted from S4 ρ18 3 3 3 -1 3 -1 -1 -1 3 0 0 -1 -1 1 1 -1 -1 1 1 3 -1 -1 0 0 0 0 -1 1 -1 1 orthogonal lifted from S4 ρ19 3 3 -3 -1 -3 -1 1 1 3 0 0 1 1 -1 -1 -1 -1 1 1 3 -1 -1 0 0 0 0 1 -1 1 -1 orthogonal lifted from C2×S4 ρ20 3 3 -3 -1 -3 -1 1 1 3 0 0 -1 -1 1 1 1 1 -1 -1 3 -1 -1 0 0 0 0 -1 1 -1 1 orthogonal lifted from C2×S4 ρ21 3 -3 -3 1 3 -1 1 -1 3 0 0 -i i -i i i -i -i i -3 -1 1 0 0 0 0 i i -i -i complex lifted from A4⋊C4 ρ22 3 -3 -3 1 3 -1 1 -1 3 0 0 i -i i -i -i i i -i -3 -1 1 0 0 0 0 -i -i i i complex lifted from A4⋊C4 ρ23 3 -3 3 1 -3 -1 -1 1 3 0 0 i -i i -i i -i -i i -3 -1 1 0 0 0 0 -i -i i i complex lifted from A4⋊C4 ρ24 3 -3 3 1 -3 -1 -1 1 3 0 0 -i i -i i -i i i -i -3 -1 1 0 0 0 0 i i -i -i complex lifted from A4⋊C4 ρ25 4 4 0 4 0 4 0 0 -2 -2 1 0 0 0 0 0 0 0 0 -2 -2 -2 -2 1 0 0 0 0 0 0 orthogonal lifted from S32 ρ26 4 -4 0 -4 0 4 0 0 -2 -2 1 0 0 0 0 0 0 0 0 2 -2 2 2 -1 0 0 0 0 0 0 symplectic lifted from S3×Dic3, Schur index 2 ρ27 6 6 0 -2 0 -2 0 0 -3 0 0 -2 -2 2 2 0 0 0 0 -3 1 1 0 0 0 0 1 -1 1 -1 orthogonal lifted from S3×S4 ρ28 6 6 0 -2 0 -2 0 0 -3 0 0 2 2 -2 -2 0 0 0 0 -3 1 1 0 0 0 0 -1 1 -1 1 orthogonal lifted from S3×S4 ρ29 6 -6 0 2 0 -2 0 0 -3 0 0 -2i 2i -2i 2i 0 0 0 0 3 1 -1 0 0 0 0 -i -i i i complex faithful ρ30 6 -6 0 2 0 -2 0 0 -3 0 0 2i -2i 2i -2i 0 0 0 0 3 1 -1 0 0 0 0 i i -i -i complex faithful

Smallest permutation representation of S3×A4⋊C4
On 36 points
Generators in S36
(1 6 10)(2 7 11)(3 8 12)(4 5 9)(13 27 17)(14 28 18)(15 25 19)(16 26 20)(21 35 31)(22 36 32)(23 33 29)(24 34 30)
(5 9)(6 10)(7 11)(8 12)(17 27)(18 28)(19 25)(20 26)(21 35)(22 36)(23 33)(24 34)
(1 2)(3 4)(5 8)(6 7)(9 12)(10 11)(13 15)(14 31)(16 29)(17 19)(18 35)(20 33)(21 28)(22 24)(23 26)(25 27)(30 32)(34 36)
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 32)(14 29)(15 30)(16 31)(17 36)(18 33)(19 34)(20 35)(21 26)(22 27)(23 28)(24 25)
(1 16 30)(2 31 13)(3 14 32)(4 29 15)(5 23 25)(6 26 24)(7 21 27)(8 28 22)(9 33 19)(10 20 34)(11 35 17)(12 18 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)

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

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

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

Matrix representation of S3×A4⋊C4 in GL5(𝔽13)

 11 11 0 0 0 8 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 5 12 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 12 0 0 0 0 8 5 12
,
 1 0 0 0 0 0 1 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 5 8 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 3 2 8 0 0 11 10 8 0 0 1 0 0
,
 12 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 12 0 0 0 0 11 10 8

G:=sub<GL(5,GF(13))| [11,8,0,0,0,11,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,5,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,0,12,8,0,0,12,0,5,0,0,0,0,12],[1,0,0,0,0,0,1,0,0,0,0,0,12,0,5,0,0,0,12,8,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,3,11,1,0,0,2,10,0,0,0,8,8,0],[12,0,0,0,0,0,12,0,0,0,0,0,0,12,11,0,0,1,0,10,0,0,0,0,8] >;

S3×A4⋊C4 in GAP, Magma, Sage, TeX

S_3\times A_4\rtimes C_4
% in TeX

G:=Group("S3xA4:C4");
// GroupNames label

G:=SmallGroup(288,856);
// by ID

G=gap.SmallGroup(288,856);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-3,-2,2,36,234,1684,3036,782,1777,1350]);
// Polycyclic

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

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