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## G = S3×C4.A4order 288 = 25·32

### Direct product of S3 and C4.A4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C3×Q8 — S3×C4.A4
 Chief series C1 — C2 — C6 — C3×Q8 — C3×SL2(𝔽3) — S3×SL2(𝔽3) — S3×C4.A4
 Lower central C3×Q8 — S3×C4.A4
 Upper central C1 — C4

Generators and relations for S3×C4.A4
G = < a,b,c,d,e,f | a3=b2=c4=f3=1, d2=e2=c2, bab=a-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, ede-1=c2d, fdf-1=c2de, fef-1=d >

Subgroups: 478 in 103 conjugacy classes, 25 normal (23 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, S3, C6, C6, C2×C4, D4, Q8, Q8, C23, C32, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C22×C4, C2×D4, C2×Q8, C4○D4, C4○D4, C3×S3, C3×C6, SL2(𝔽3), SL2(𝔽3), Dic6, C4×S3, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C3×Q8, C22×S3, C2×C4○D4, C3×Dic3, C3×C12, S3×C6, C2×SL2(𝔽3), C4.A4, C4.A4, S3×C2×C4, C4○D12, S3×D4, D42S3, S3×Q8, Q83S3, C3×C4○D4, C3×SL2(𝔽3), S3×C12, C2×C4.A4, S3×C4○D4, Dic3.A4, S3×SL2(𝔽3), C3×C4.A4, S3×C4.A4
Quotients: C1, C2, C3, C22, S3, C6, A4, D6, C2×C6, C3×S3, C2×A4, S3×C6, C4.A4, C22×A4, S3×A4, C2×C4.A4, C2×S3×A4, S3×C4.A4

Smallest permutation representation of S3×C4.A4
On 48 points
Generators in S48
(1 39 9)(2 40 10)(3 37 11)(4 38 12)(5 16 22)(6 13 23)(7 14 24)(8 15 21)(17 35 41)(18 36 42)(19 33 43)(20 34 44)(25 47 31)(26 48 32)(27 45 29)(28 46 30)
(1 3)(2 4)(5 24)(6 21)(7 22)(8 23)(9 37)(10 38)(11 39)(12 40)(13 15)(14 16)(17 19)(18 20)(25 29)(26 30)(27 31)(28 32)(33 41)(34 42)(35 43)(36 44)(45 47)(46 48)
(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)(37 38 39 40)(41 42 43 44)(45 46 47 48)
(1 19 3 17)(2 20 4 18)(5 26 7 28)(6 27 8 25)(9 43 11 41)(10 44 12 42)(13 45 15 47)(14 46 16 48)(21 31 23 29)(22 32 24 30)(33 37 35 39)(34 38 36 40)
(1 47 3 45)(2 48 4 46)(5 42 7 44)(6 43 8 41)(9 25 11 27)(10 26 12 28)(13 19 15 17)(14 20 16 18)(21 35 23 33)(22 36 24 34)(29 39 31 37)(30 40 32 38)
(5 42 28)(6 43 25)(7 44 26)(8 41 27)(13 19 47)(14 20 48)(15 17 45)(16 18 46)(21 35 29)(22 36 30)(23 33 31)(24 34 32)

G:=sub<Sym(48)| (1,39,9)(2,40,10)(3,37,11)(4,38,12)(5,16,22)(6,13,23)(7,14,24)(8,15,21)(17,35,41)(18,36,42)(19,33,43)(20,34,44)(25,47,31)(26,48,32)(27,45,29)(28,46,30), (1,3)(2,4)(5,24)(6,21)(7,22)(8,23)(9,37)(10,38)(11,39)(12,40)(13,15)(14,16)(17,19)(18,20)(25,29)(26,30)(27,31)(28,32)(33,41)(34,42)(35,43)(36,44)(45,47)(46,48), (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)(37,38,39,40)(41,42,43,44)(45,46,47,48), (1,19,3,17)(2,20,4,18)(5,26,7,28)(6,27,8,25)(9,43,11,41)(10,44,12,42)(13,45,15,47)(14,46,16,48)(21,31,23,29)(22,32,24,30)(33,37,35,39)(34,38,36,40), (1,47,3,45)(2,48,4,46)(5,42,7,44)(6,43,8,41)(9,25,11,27)(10,26,12,28)(13,19,15,17)(14,20,16,18)(21,35,23,33)(22,36,24,34)(29,39,31,37)(30,40,32,38), (5,42,28)(6,43,25)(7,44,26)(8,41,27)(13,19,47)(14,20,48)(15,17,45)(16,18,46)(21,35,29)(22,36,30)(23,33,31)(24,34,32)>;

G:=Group( (1,39,9)(2,40,10)(3,37,11)(4,38,12)(5,16,22)(6,13,23)(7,14,24)(8,15,21)(17,35,41)(18,36,42)(19,33,43)(20,34,44)(25,47,31)(26,48,32)(27,45,29)(28,46,30), (1,3)(2,4)(5,24)(6,21)(7,22)(8,23)(9,37)(10,38)(11,39)(12,40)(13,15)(14,16)(17,19)(18,20)(25,29)(26,30)(27,31)(28,32)(33,41)(34,42)(35,43)(36,44)(45,47)(46,48), (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)(37,38,39,40)(41,42,43,44)(45,46,47,48), (1,19,3,17)(2,20,4,18)(5,26,7,28)(6,27,8,25)(9,43,11,41)(10,44,12,42)(13,45,15,47)(14,46,16,48)(21,31,23,29)(22,32,24,30)(33,37,35,39)(34,38,36,40), (1,47,3,45)(2,48,4,46)(5,42,7,44)(6,43,8,41)(9,25,11,27)(10,26,12,28)(13,19,15,17)(14,20,16,18)(21,35,23,33)(22,36,24,34)(29,39,31,37)(30,40,32,38), (5,42,28)(6,43,25)(7,44,26)(8,41,27)(13,19,47)(14,20,48)(15,17,45)(16,18,46)(21,35,29)(22,36,30)(23,33,31)(24,34,32) );

G=PermutationGroup([[(1,39,9),(2,40,10),(3,37,11),(4,38,12),(5,16,22),(6,13,23),(7,14,24),(8,15,21),(17,35,41),(18,36,42),(19,33,43),(20,34,44),(25,47,31),(26,48,32),(27,45,29),(28,46,30)], [(1,3),(2,4),(5,24),(6,21),(7,22),(8,23),(9,37),(10,38),(11,39),(12,40),(13,15),(14,16),(17,19),(18,20),(25,29),(26,30),(27,31),(28,32),(33,41),(34,42),(35,43),(36,44),(45,47),(46,48)], [(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),(37,38,39,40),(41,42,43,44),(45,46,47,48)], [(1,19,3,17),(2,20,4,18),(5,26,7,28),(6,27,8,25),(9,43,11,41),(10,44,12,42),(13,45,15,47),(14,46,16,48),(21,31,23,29),(22,32,24,30),(33,37,35,39),(34,38,36,40)], [(1,47,3,45),(2,48,4,46),(5,42,7,44),(6,43,8,41),(9,25,11,27),(10,26,12,28),(13,19,15,17),(14,20,16,18),(21,35,23,33),(22,36,24,34),(29,39,31,37),(30,40,32,38)], [(5,42,28),(6,43,25),(7,44,26),(8,41,27),(13,19,47),(14,20,48),(15,17,45),(16,18,46),(21,35,29),(22,36,30),(23,33,31),(24,34,32)]])

42 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 3C 3D 3E 4A 4B 4C 4D 4E 4F 6A 6B 6C 6D 6E 6F ··· 6J 12A 12B 12C 12D 12E 12F 12G 12H 12I 12J 12K ··· 12O order 1 2 2 2 2 2 3 3 3 3 3 4 4 4 4 4 4 6 6 6 6 6 6 ··· 6 12 12 12 12 12 12 12 12 12 12 12 ··· 12 size 1 1 3 3 6 18 2 4 4 8 8 1 1 3 3 6 18 2 4 4 8 8 12 ··· 12 2 2 4 4 4 4 8 8 8 8 12 ··· 12

42 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 3 3 3 3 4 6 6 type + + + + + + + + + + + + image C1 C2 C2 C2 C3 C6 C6 C6 S3 D6 C3×S3 S3×C6 C4.A4 A4 C2×A4 C2×A4 C2×A4 S3×C4.A4 S3×A4 C2×S3×A4 kernel S3×C4.A4 Dic3.A4 S3×SL2(𝔽3) C3×C4.A4 S3×C4○D4 S3×Q8 Q8⋊3S3 C3×C4○D4 C4.A4 SL2(𝔽3) C4○D4 Q8 S3 C4×S3 Dic3 C12 D6 C1 C4 C2 # reps 1 1 1 1 2 2 2 2 1 1 2 2 12 1 1 1 1 6 1 1

Matrix representation of S3×C4.A4 in GL4(𝔽5) generated by

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

S3×C4.A4 in GAP, Magma, Sage, TeX

S_3\times C_4.A_4
% in TeX

G:=Group("S3xC4.A4");
// GroupNames label

G:=SmallGroup(288,925);
// by ID

G=gap.SmallGroup(288,925);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,2,-3,-2,1016,269,360,123,515,242,4037]);
// Polycyclic

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

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