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## G = C2×S3×SL2(𝔽3)  order 288 = 25·32

### Direct product of C2, S3 and SL2(𝔽3)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C3×Q8 — C2×S3×SL2(𝔽3)
 Chief series C1 — C2 — C6 — C3×Q8 — C3×SL2(𝔽3) — S3×SL2(𝔽3) — C2×S3×SL2(𝔽3)
 Lower central C3×Q8 — C2×S3×SL2(𝔽3)
 Upper central C1 — C22

Generators and relations for C2×S3×SL2(𝔽3)
G = < a,b,c,d,e,f | a2=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: 550 in 129 conjugacy classes, 35 normal (17 characteristic)
C1, C2, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C6, C2×C4, Q8, Q8, C23, C32, Dic3, C12, D6, C2×C6, C2×C6, C22×C4, C2×Q8, C2×Q8, C3×S3, C3×C6, SL2(𝔽3), SL2(𝔽3), Dic6, C4×S3, C2×Dic3, C2×C12, C3×Q8, C3×Q8, C22×S3, C22×C6, C22×Q8, S3×C6, C62, C2×SL2(𝔽3), C2×SL2(𝔽3), C2×Dic6, S3×C2×C4, S3×Q8, S3×Q8, C6×Q8, C3×SL2(𝔽3), S3×C2×C6, C22×SL2(𝔽3), C2×S3×Q8, S3×SL2(𝔽3), C6×SL2(𝔽3), C2×S3×SL2(𝔽3)
Quotients: C1, C2, C3, C22, S3, C6, A4, D6, C2×C6, C3×S3, SL2(𝔽3), C2×A4, S3×C6, C2×SL2(𝔽3), C22×A4, S3×A4, C22×SL2(𝔽3), S3×SL2(𝔽3), C2×S3×A4, C2×S3×SL2(𝔽3)

Smallest permutation representation of C2×S3×SL2(𝔽3)
On 48 points
Generators in S48
(1 33)(2 34)(3 35)(4 36)(5 25)(6 26)(7 27)(8 28)(9 30)(10 31)(11 32)(12 29)(13 39)(14 40)(15 37)(16 38)(17 43)(18 44)(19 41)(20 42)(21 47)(22 48)(23 45)(24 46)
(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 35)(2 36)(3 33)(4 34)(5 19)(6 20)(7 17)(8 18)(9 40)(10 37)(11 38)(12 39)(13 29)(14 30)(15 31)(16 32)(21 45)(22 46)(23 47)(24 48)(25 41)(26 42)(27 43)(28 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 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)
(2 23 24)(4 21 22)(5 6 39)(7 8 37)(10 17 18)(12 19 20)(13 25 26)(15 27 28)(29 41 42)(31 43 44)(34 45 46)(36 47 48)

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

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

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

42 conjugacy classes

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

42 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 3 3 3 4 4 6 6 type + + + + + - + + + - + + image C1 C2 C2 C3 C6 C6 S3 D6 C3×S3 SL2(𝔽3) SL2(𝔽3) S3×C6 A4 C2×A4 C2×A4 S3×SL2(𝔽3) S3×SL2(𝔽3) S3×A4 C2×S3×A4 kernel C2×S3×SL2(𝔽3) S3×SL2(𝔽3) C6×SL2(𝔽3) C2×S3×Q8 S3×Q8 C6×Q8 C2×SL2(𝔽3) SL2(𝔽3) C2×Q8 D6 D6 Q8 C22×S3 D6 C2×C6 C2 C2 C22 C2 # reps 1 2 1 2 4 2 1 1 2 4 8 2 1 2 1 2 4 1 1

Matrix representation of C2×S3×SL2(𝔽3) in GL4(𝔽13) generated by

 12 0 0 0 0 12 0 0 0 0 1 0 0 0 0 1
,
 12 12 0 0 1 0 0 0 0 0 1 0 0 0 0 1
,
 12 0 0 0 1 1 0 0 0 0 12 0 0 0 0 12
,
 1 0 0 0 0 1 0 0 0 0 1 11 0 0 1 12
,
 1 0 0 0 0 1 0 0 0 0 12 7 0 0 9 1
,
 3 0 0 0 0 3 0 0 0 0 1 0 0 0 1 3
G:=sub<GL(4,GF(13))| [12,0,0,0,0,12,0,0,0,0,1,0,0,0,0,1],[12,1,0,0,12,0,0,0,0,0,1,0,0,0,0,1],[12,1,0,0,0,1,0,0,0,0,12,0,0,0,0,12],[1,0,0,0,0,1,0,0,0,0,1,1,0,0,11,12],[1,0,0,0,0,1,0,0,0,0,12,9,0,0,7,1],[3,0,0,0,0,3,0,0,0,0,1,1,0,0,0,3] >;

C2×S3×SL2(𝔽3) in GAP, Magma, Sage, TeX

C_2\times S_3\times {\rm SL}_2({\mathbb F}_3)
% in TeX

G:=Group("C2xS3xSL(2,3)");
// GroupNames label

G:=SmallGroup(288,922);
// by ID

G=gap.SmallGroup(288,922);
# by ID

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

G:=Group<a,b,c,d,e,f|a^2=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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