direct product, non-abelian, soluble
Aliases: C3×S3×SL2(𝔽3), D6.(C3×A4), C6.2(C6×A4), (S3×Q8)⋊C32, (S3×C6).3A4, C6.21(S3×A4), C3⋊(C6×SL2(𝔽3)), (Q8×C32)⋊5C6, Q8⋊2(S3×C32), (C3×SL2(𝔽3))⋊6C6, C32⋊4(C2×SL2(𝔽3)), (C32×SL2(𝔽3))⋊4C2, (C3×S3×Q8)⋊C3, C2.3(C3×S3×A4), (C3×Q8)⋊(C3×C6), (C3×Q8)⋊4(C3×S3), (C3×C6).16(C2×A4), SmallGroup(432,623)
Series: Derived ►Chief ►Lower central ►Upper central
| C3×Q8 — C3×S3×SL2(𝔽3) |
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)
►On 48 points
(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