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## G = C3×C4.3S4order 288 = 25·32

### Direct product of C3 and C4.3S4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8 — SL2(𝔽3) — C3×C4.3S4
 Chief series C1 — C2 — Q8 — SL2(𝔽3) — C3×SL2(𝔽3) — C3×GL2(𝔽3) — C3×C4.3S4
 Lower central SL2(𝔽3) — C3×C4.3S4
 Upper central C1 — C6 — C12

Generators and relations for C3×C4.3S4
G = < a,b,c,d,e,f | a3=b4=e3=f2=1, c2=d2=b2, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, fbf=b-1, dcd-1=b2c, ece-1=b2cd, fcf=cd, ede-1=c, fdf=b2d, fef=e-1 >

Subgroups: 374 in 89 conjugacy classes, 20 normal (16 characteristic)
C1, C2, C2 [×3], C3, C3 [×2], C4, C4, C22 [×5], S3 [×2], C6, C6 [×5], C8 [×2], C2×C4, D4 [×4], Q8, C23, C32, C12, C12 [×3], D6 [×2], C2×C6 [×5], M4(2), D8 [×2], SD16 [×2], C2×D4, C4○D4, C3×S3 [×2], C3×C6, C24 [×2], SL2(𝔽3), SL2(𝔽3), D12, C2×C12, C3×D4 [×4], C3×Q8, C22×C6, C8⋊C22, C3×C12, S3×C6 [×2], C3×M4(2), C3×D8 [×2], C3×SD16 [×2], GL2(𝔽3) [×2], C4.A4, C4.A4, C6×D4, C3×C4○D4, C3×SL2(𝔽3), C3×D12, C3×C8⋊C22, C4.3S4, C3×GL2(𝔽3) [×2], C3×C4.A4, C3×C4.3S4
Quotients: C1, C2 [×3], C3, C22, S3, C6 [×3], D6, C2×C6, C3×S3, S4, S3×C6, C2×S4, C3×S4, C4.3S4, C6×S4, C3×C4.3S4

Smallest permutation representation of C3×C4.3S4
On 48 points
Generators in S48
(1 31 35)(2 32 36)(3 29 33)(4 30 34)(5 14 10)(6 15 11)(7 16 12)(8 13 9)(17 27 21)(18 28 22)(19 25 23)(20 26 24)(37 45 41)(38 46 42)(39 47 43)(40 48 44)
(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 37 3 39)(2 38 4 40)(5 28 7 26)(6 25 8 27)(9 17 11 19)(10 18 12 20)(13 21 15 23)(14 22 16 24)(29 47 31 45)(30 48 32 46)(33 43 35 41)(34 44 36 42)
(1 17 3 19)(2 18 4 20)(5 48 7 46)(6 45 8 47)(9 39 11 37)(10 40 12 38)(13 43 15 41)(14 44 16 42)(21 33 23 35)(22 34 24 36)(25 31 27 29)(26 32 28 30)
(5 48 26)(6 45 27)(7 46 28)(8 47 25)(9 39 19)(10 40 20)(11 37 17)(12 38 18)(13 43 23)(14 44 24)(15 41 21)(16 42 22)
(2 4)(5 48)(6 47)(7 46)(8 45)(9 37)(10 40)(11 39)(12 38)(13 41)(14 44)(15 43)(16 42)(17 19)(21 23)(25 27)(30 32)(34 36)

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

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

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

39 conjugacy classes

 class 1 2A 2B 2C 2D 3A 3B 3C 3D 3E 4A 4B 6A 6B 6C 6D 6E 6F 6G 6H 6I 6J 6K 8A 8B 12A 12B 12C 12D 12E ··· 12J 24A 24B 24C 24D order 1 2 2 2 2 3 3 3 3 3 4 4 6 6 6 6 6 6 6 6 6 6 6 8 8 12 12 12 12 12 ··· 12 24 24 24 24 size 1 1 6 12 12 1 1 8 8 8 2 6 1 1 6 6 8 8 8 12 12 12 12 12 12 2 2 6 6 8 ··· 8 12 12 12 12

39 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 3 3 3 3 4 4 type + + + + + + + + image C1 C2 C2 C3 C6 C6 S3 D6 C3×S3 S3×C6 S4 C2×S4 C3×S4 C6×S4 C4.3S4 C3×C4.3S4 kernel C3×C4.3S4 C3×GL2(𝔽3) C3×C4.A4 C4.3S4 GL2(𝔽3) C4.A4 C3×C4○D4 C3×Q8 C4○D4 Q8 C12 C6 C4 C2 C3 C1 # reps 1 2 1 2 4 2 1 1 2 2 2 2 4 4 3 6

Matrix representation of C3×C4.3S4 in GL4(𝔽7) generated by

 4 0 0 0 0 4 0 0 0 0 4 0 0 0 0 4
,
 2 6 6 6 0 5 2 5 6 4 6 6 6 3 4 1
,
 5 1 6 4 0 2 2 3 2 2 6 6 1 4 4 1
,
 0 3 4 1 2 6 0 4 6 3 4 3 4 5 5 4
,
 5 0 5 4 6 0 5 1 6 3 2 1 4 5 4 5
,
 2 6 6 6 4 3 0 1 5 6 1 3 1 3 4 1
G:=sub<GL(4,GF(7))| [4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[2,0,6,6,6,5,4,3,6,2,6,4,6,5,6,1],[5,0,2,1,1,2,2,4,6,2,6,4,4,3,6,1],[0,2,6,4,3,6,3,5,4,0,4,5,1,4,3,4],[5,6,6,4,0,0,3,5,5,5,2,4,4,1,1,5],[2,4,5,1,6,3,6,3,6,0,1,4,6,1,3,1] >;

C3×C4.3S4 in GAP, Magma, Sage, TeX

C_3\times C_4._3S_4
% in TeX

G:=Group("C3xC4.3S4");
// GroupNames label

G:=SmallGroup(288,904);
// by ID

G=gap.SmallGroup(288,904);
# by ID

G:=PCGroup([7,-2,-2,-3,-3,-2,2,-2,2045,1016,675,2524,655,172,1517,404,285,124]);
// Polycyclic

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

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