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## G = C3×C4.6S4order 288 = 25·32

### Direct product of C3 and C4.6S4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8 — SL2(𝔽3) — C3×C4.6S4
 Chief series C1 — C2 — Q8 — SL2(𝔽3) — C3×SL2(𝔽3) — C3×GL2(𝔽3) — C3×C4.6S4
 Lower central SL2(𝔽3) — C3×C4.6S4
 Upper central C1 — C12

Generators and relations for C3×C4.6S4
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, bf=fb, dcd-1=b2c, ece-1=b2cd, fcf=cd, ede-1=c, fdf=b2d, fef=e-1 >

Subgroups: 286 in 83 conjugacy classes, 20 normal (all characteristic)
C1, C2, C2 [×2], C3, C3 [×2], C4, C4 [×2], C22 [×2], S3 [×2], C6, C6 [×4], C8 [×2], C2×C4 [×2], D4 [×3], Q8, Q8, C32, Dic3, C12, C12 [×4], D6, C2×C6 [×2], C2×C8, D8, SD16 [×2], Q16, C4○D4, C4○D4, C3×S3 [×2], C3×C6, C24 [×2], SL2(𝔽3), SL2(𝔽3), C4×S3, C2×C12 [×2], C3×D4 [×3], C3×Q8, C3×Q8, C4○D8, C3×Dic3, C3×C12, S3×C6, C2×C24, C3×D8, C3×SD16 [×2], C3×Q16, CSU2(𝔽3), GL2(𝔽3), C4.A4, C4.A4, C3×C4○D4, C3×C4○D4, C3×SL2(𝔽3), S3×C12, C3×C4○D8, C4.6S4, C3×CSU2(𝔽3), C3×GL2(𝔽3), C3×C4.A4, C3×C4.6S4
Quotients: C1, C2 [×3], C3, C22, S3, C6 [×3], D6, C2×C6, C3×S3, S4, S3×C6, C2×S4, C3×S4, C4.6S4, C6×S4, C3×C4.6S4

Smallest permutation representation of C3×C4.6S4
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)
(1 3)(2 4)(5 48)(6 45)(7 46)(8 47)(9 39)(10 40)(11 37)(12 38)(13 43)(14 44)(15 41)(16 42)(29 31)(30 32)(33 35)(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), (1,3)(2,4)(5,48)(6,45)(7,46)(8,47)(9,39)(10,40)(11,37)(12,38)(13,43)(14,44)(15,41)(16,42)(29,31)(30,32)(33,35)(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), (1,3)(2,4)(5,48)(6,45)(7,46)(8,47)(9,39)(10,40)(11,37)(12,38)(13,43)(14,44)(15,41)(16,42)(29,31)(30,32)(33,35)(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)], [(1,3),(2,4),(5,48),(6,45),(7,46),(8,47),(9,39),(10,40),(11,37),(12,38),(13,43),(14,44),(15,41),(16,42),(29,31),(30,32),(33,35),(34,36)])

48 conjugacy classes

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

48 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 3 3 3 3 4 4 type + + + + + + + + image C1 C2 C2 C2 C3 C6 C6 C6 S3 D6 C3×S3 S3×C6 C4.6S4 C3×C4.6S4 S4 C2×S4 C3×S4 C6×S4 C4.6S4 C3×C4.6S4 kernel C3×C4.6S4 C3×CSU2(𝔽3) C3×GL2(𝔽3) C3×C4.A4 C4.6S4 CSU2(𝔽3) GL2(𝔽3) C4.A4 C3×C4○D4 C3×Q8 C4○D4 Q8 C3 C1 C12 C6 C4 C2 C3 C1 # reps 1 1 1 1 2 2 2 2 1 1 2 2 4 8 2 2 4 4 2 4

Matrix representation of C3×C4.6S4 in GL2(𝔽73) generated by

 8 0 0 8
,
 46 0 0 46
,
 16 29 44 57
,
 28 29 56 45
,
 0 72 1 72
,
 72 1 0 1
G:=sub<GL(2,GF(73))| [8,0,0,8],[46,0,0,46],[16,44,29,57],[28,56,29,45],[0,1,72,72],[72,0,1,1] >;

C3×C4.6S4 in GAP, Magma, Sage, TeX

C_3\times C_4._6S_4
% in TeX

G:=Group("C3xC4.6S4");
// GroupNames label

G:=SmallGroup(288,903);
// by ID

G=gap.SmallGroup(288,903);
# by ID

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