direct product, non-abelian, soluble
Aliases: C6×GL2(𝔽3), Q8⋊(S3×C6), C2.6(C6×S4), (C6×Q8)⋊3S3, (C3×Q8)⋊3D6, C6.43(C2×S4), (C2×C6).20S4, C22.5(C3×S4), (C6×SL2(𝔽3))⋊6C2, (C2×SL2(𝔽3))⋊2C6, SL2(𝔽3)⋊1(C2×C6), (C3×SL2(𝔽3))⋊9C22, (C2×Q8)⋊1(C3×S3), SmallGroup(288,900)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2 — Q8 — SL2(𝔽3) — C3×SL2(𝔽3) — C3×GL2(𝔽3) — C6×GL2(𝔽3) |
| SL2(𝔽3) — C6×GL2(𝔽3) |
Generators and relations for C6×GL2(𝔽3)
G = < a,b,c,d,e | a6=b4=d3=e2=1, c2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ece=b-1, dbd-1=bc, ebe=b2c, dcd-1=b, ede=d-1 >
Subgroups: 430 in 109 conjugacy classes, 24 normal (16 characteristic)
C1, C2, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C6, C8, C2×C4, D4, Q8, Q8, C23, C32, C12, D6, C2×C6, C2×C6, C2×C8, SD16, C2×D4, C2×Q8, C3×S3, C3×C6, C24, SL2(𝔽3), SL2(𝔽3), C2×C12, C3×D4, C3×Q8, C3×Q8, C22×S3, C22×C6, C2×SD16, S3×C6, C62, C2×C24, C3×SD16, GL2(𝔽3), C2×SL2(𝔽3), C2×SL2(𝔽3), C6×D4, C6×Q8, C3×SL2(𝔽3), S3×C2×C6, C6×SD16, C2×GL2(𝔽3), C3×GL2(𝔽3), C6×SL2(𝔽3), C6×GL2(𝔽3)
Quotients: C1, C2, C3, C22, S3, C6, D6, C2×C6, C3×S3, S4, S3×C6, GL2(𝔽3), C2×S4, C3×S4, C2×GL2(𝔽3), C3×GL2(𝔽3), C6×S4, C6×GL2(𝔽3)
►On 48 points
(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 47 8 37)(2 48 9 38)(3 43 10 39)(4 44 11 40)(5 45 12 41)(6 46 7 42)(13 27 22 32)(14 28 23 33)(15 29 24 34)(16 30 19 35)(17 25 20 36)(18 26 21 31)
(1 16 8 19)(2 17 9 20)(3 18 10 21)(4 13 11 22)(5 14 12 23)(6 15 7 24)(25 48 36 38)(26 43 31 39)(27 44 32 40)(28 45 33 41)(29 46 34 42)(30 47 35 37)
(13 32 44)(14 33 45)(15 34 46)(16 35 47)(17 36 48)(18 31 43)(19 30 37)(20 25 38)(21 26 39)(22 27 40)(23 28 41)(24 29 42)
(13 40)(14 41)(15 42)(16 37)(17 38)(18 39)(19 47)(20 48)(21 43)(22 44)(23 45)(24 46)(25 36)(26 31)(27 32)(28 33)(29 34)(30 35)
G:=sub<Sym(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), (1,47,8,37)(2,48,9,38)(3,43,10,39)(4,44,11,40)(5,45,12,41)(6,46,7,42)(13,27,22,32)(14,28,23,33)(15,29,24,34)(16,30,19,35)(17,25,20,36)(18,26,21,31), (1,16,8,19)(2,17,9,20)(3,18,10,21)(4,13,11,22)(5,14,12,23)(6,15,7,24)(25,48,36,38)(26,43,31,39)(27,44,32,40)(28,45,33,41)(29,46,34,42)(30,47,35,37), (13,32,44)(14,33,45)(15,34,46)(16,35,47)(17,36,48)(18,31,43)(19,30,37)(20,25,38)(21,26,39)(22,27,40)(23,28,41)(24,29,42), (13,40)(14,41)(15,42)(16,37)(17,38)(18,39)(19,47)(20,48)(21,43)(22,44)(23,45)(24,46)(25,36)(26,31)(27,32)(28,33)(29,34)(30,35)>;
G:=Group( (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), (1,47,8,37)(2,48,9,38)(3,43,10,39)(4,44,11,40)(5,45,12,41)(6,46,7,42)(13,27,22,32)(14,28,23,33)(15,29,24,34)(16,30,19,35)(17,25,20,36)(18,26,21,31), (1,16,8,19)(2,17,9,20)(3,18,10,21)(4,13,11,22)(5,14,12,23)(6,15,7,24)(25,48,36,38)(26,43,31,39)(27,44,32,40)(28,45,33,41)(29,46,34,42)(30,47,35,37), (13,32,44)(14,33,45)(15,34,46)(16,35,47)(17,36,48)(18,31,43)(19,30,37)(20,25,38)(21,26,39)(22,27,40)(23,28,41)(24,29,42), (13,40)(14,41)(15,42)(16,37)(17,38)(18,39)(19,47)(20,48)(21,43)(22,44)(23,45)(24,46)(25,36)(26,31)(27,32)(28,33)(29,34)(30,35) );
G=PermutationGroup([[(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)], [(1,47,8,37),(2,48,9,38),(3,43,10,39),(4,44,11,40),(5,45,12,41),(6,46,7,42),(13,27,22,32),(14,28,23,33),(15,29,24,34),(16,30,19,35),(17,25,20,36),(18,26,21,31)], [(1,16,8,19),(2,17,9,20),(3,18,10,21),(4,13,11,22),(5,14,12,23),(6,15,7,24),(25,48,36,38),(26,43,31,39),(27,44,32,40),(28,45,33,41),(29,46,34,42),(30,47,35,37)], [(13,32,44),(14,33,45),(15,34,46),(16,35,47),(17,36,48),(18,31,43),(19,30,37),(20,25,38),(21,26,39),(22,27,40),(23,28,41),(24,29,42)], [(13,40),(14,41),(15,42),(16,37),(17,38),(18,39),(19,47),(20,48),(21,43),(22,44),(23,45),(24,46),(25,36),(26,31),(27,32),(28,33),(29,34),(30,35)]])
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 6A | ··· | 6F | 6G | ··· | 6O | 6P | 6Q | 6R | 6S | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 12 | 12 | 1 | 1 | 8 | 8 | 8 | 6 | 6 | 1 | ··· | 1 | 8 | ··· | 8 | 12 | 12 | 12 | 12 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | ··· | 6 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | ||||||||||
| image | C1 | C2 | C2 | C3 | C6 | C6 | S3 | D6 | C3×S3 | S3×C6 | GL2(𝔽3) | C3×GL2(𝔽3) | S4 | C2×S4 | C3×S4 | C6×S4 | GL2(𝔽3) | C3×GL2(𝔽3) |
| kernel | C6×GL2(𝔽3) | C3×GL2(𝔽3) | C6×SL2(𝔽3) | C2×GL2(𝔽3) | GL2(𝔽3) | C2×SL2(𝔽3) | C6×Q8 | C3×Q8 | C2×Q8 | Q8 | C6 | C2 | C2×C6 | C6 | C22 | C2 | C6 | C2 |
| # reps | 1 | 2 | 1 | 2 | 4 | 2 | 1 | 1 | 2 | 2 | 4 | 8 | 2 | 2 | 4 | 4 | 2 | 4 |
Matrix representation of C6×GL2(𝔽3) ►in GL3(𝔽73) generated by
| 72 | 0 | 0 |
| 0 | 65 | 0 |
| 0 | 0 | 65 |
| 1 | 0 | 0 |
| 0 | 32 | 52 |
| 0 | 21 | 41 |
| 1 | 0 | 0 |
| 0 | 20 | 52 |
| 0 | 33 | 53 |
| 1 | 0 | 0 |
| 0 | 52 | 32 |
| 0 | 53 | 20 |
| 72 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 72 | 72 |
G:=sub<GL(3,GF(73))| [72,0,0,0,65,0,0,0,65],[1,0,0,0,32,21,0,52,41],[1,0,0,0,20,33,0,52,53],[1,0,0,0,52,53,0,32,20],[72,0,0,0,1,72,0,0,72] >;
C6×GL2(𝔽3) in GAP, Magma, Sage, TeX
C_6\times {\rm GL}_2({\mathbb F}_3) % in TeX
G:=Group("C6xGL(2,3)"); // GroupNames label
G:=SmallGroup(288,900);
// by ID
G=gap.SmallGroup(288,900);
# by ID
G:=PCGroup([7,-2,-2,-3,-3,-2,2,-2,675,2524,655,172,1517,404,285,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^6=b^4=d^3=e^2=1,c^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=e*c*e=b^-1,d*b*d^-1=b*c,e*b*e=b^2*c,d*c*d^-1=b,e*d*e=d^-1>;
// generators/relations