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## G = C3×S3×SL2(𝔽3)  order 432 = 24·33

### Direct product of C3, S3 and SL2(𝔽3)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C3×Q8 — C3×S3×SL2(𝔽3)
 Chief series C1 — C2 — C6 — C3×Q8 — Q8×C32 — C32×SL2(𝔽3) — C3×S3×SL2(𝔽3)
 Lower central C3×Q8 — C3×S3×SL2(𝔽3)
 Upper central C1 — C6

Generators and relations for C3×S3×SL2(𝔽3)
G = < a,b,c,d,e,f | a3=b3=c2=d4=f3=1, e2=d2, ab=ba, ac=ca, ad=da, ae=ea, af=fa, cbc=b-1, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, ede-1=d-1, fdf-1=e, fef-1=de >

Subgroups: 546 in 125 conjugacy classes, 32 normal (19 characteristic)
C1, C2, C2, C3, C3, C4, C22, S3, C6, C6, C2×C4, Q8, Q8, C32, C32, Dic3, C12, D6, C2×C6, C2×Q8, C3×S3, C3×S3, C3×C6, C3×C6, SL2(𝔽3), SL2(𝔽3), Dic6, C4×S3, C2×C12, C3×Q8, C3×Q8, C33, C3×Dic3, C3×C12, S3×C6, S3×C6, C62, C2×SL2(𝔽3), S3×Q8, C6×Q8, S3×C32, C32×C6, C3×SL2(𝔽3), C3×SL2(𝔽3), C3×SL2(𝔽3), C3×Dic6, S3×C12, Q8×C32, S3×C3×C6, S3×SL2(𝔽3), C6×SL2(𝔽3), C3×S3×Q8, C32×SL2(𝔽3), C3×S3×SL2(𝔽3)
Quotients: C1, C2, C3, S3, C6, C32, A4, C3×S3, C3×C6, SL2(𝔽3), C2×A4, C3×A4, C2×SL2(𝔽3), S3×C32, C3×SL2(𝔽3), S3×A4, C6×A4, S3×SL2(𝔽3), C6×SL2(𝔽3), C3×S3×A4, C3×S3×SL2(𝔽3)

Smallest permutation representation of C3×S3×SL2(𝔽3)
On 48 points
Generators in S48
(1 9 16)(2 10 13)(3 11 14)(4 12 15)(5 45 43)(6 46 44)(7 47 41)(8 48 42)(17 25 23)(18 26 24)(19 27 21)(20 28 22)(29 37 36)(30 38 33)(31 39 34)(32 40 35)
(1 9 16)(2 10 13)(3 11 14)(4 12 15)(5 43 45)(6 44 46)(7 41 47)(8 42 48)(17 25 23)(18 26 24)(19 27 21)(20 28 22)(29 36 37)(30 33 38)(31 34 39)(32 35 40)
(1 35)(2 36)(3 33)(4 34)(5 27)(6 28)(7 25)(8 26)(9 32)(10 29)(11 30)(12 31)(13 37)(14 38)(15 39)(16 40)(17 41)(18 42)(19 43)(20 44)(21 45)(22 46)(23 47)(24 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 24 3 22)(2 23 4 21)(5 37 7 39)(6 40 8 38)(9 18 11 20)(10 17 12 19)(13 25 15 27)(14 28 16 26)(29 41 31 43)(30 44 32 42)(33 46 35 48)(34 45 36 47)
(1 16 9)(2 25 18)(3 14 11)(4 27 20)(5 44 34)(6 31 45)(7 42 36)(8 29 47)(10 23 26)(12 21 28)(13 17 24)(15 19 22)(30 33 38)(32 35 40)(37 41 48)(39 43 46)

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

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

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

63 conjugacy classes

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

63 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 3 3 3 3 4 4 4 6 6 type + + + - + + - + image C1 C2 C3 C3 C6 C6 S3 C3×S3 C3×S3 SL2(𝔽3) SL2(𝔽3) C3×SL2(𝔽3) A4 C2×A4 C3×A4 C6×A4 S3×SL2(𝔽3) S3×SL2(𝔽3) C3×S3×SL2(𝔽3) S3×A4 C3×S3×A4 kernel C3×S3×SL2(𝔽3) C32×SL2(𝔽3) S3×SL2(𝔽3) C3×S3×Q8 C3×SL2(𝔽3) Q8×C32 C3×SL2(𝔽3) SL2(𝔽3) C3×Q8 C3×S3 C3×S3 S3 S3×C6 C3×C6 D6 C6 C3 C3 C1 C6 C2 # reps 1 1 6 2 6 2 1 6 2 2 4 12 1 1 2 2 1 2 6 1 2

Matrix representation of C3×S3×SL2(𝔽3) in GL4(𝔽7) generated by

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

C3×S3×SL2(𝔽3) in GAP, Magma, Sage, TeX

C_3\times S_3\times {\rm SL}_2({\mathbb F}_3)
% in TeX

G:=Group("C3xS3xSL(2,3)");
// GroupNames label

G:=SmallGroup(432,623);
// by ID

G=gap.SmallGroup(432,623);
# by ID

G:=PCGroup([7,-2,-3,-3,-2,2,-3,-2,766,360,326,515,242,6053]);
// Polycyclic

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

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