direct product, non-abelian, soluble
Aliases: C3×C4.3S4, C12.17S4, GL2(𝔽3)⋊2C6, C4.A4⋊1C6, C4.3(C3×S4), C2.10(C6×S4), C6.47(C2×S4), Q8.5(S3×C6), (C3×Q8).23D6, (C3×GL2(𝔽3))⋊6C2, SL2(𝔽3)⋊2(C2×C6), (C3×SL2(𝔽3))⋊10C22, (C3×C4.A4)⋊6C2, (C3×C4○D4)⋊4S3, C4○D4⋊2(C3×S3), SmallGroup(288,904)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2 — Q8 — SL2(𝔽3) — C3×SL2(𝔽3) — C3×GL2(𝔽3) — C3×C4.3S4 |
| SL2(𝔽3) — C3×C4.3S4 |
Generators and relations for C3×C4.3S4
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, fbf=b-1, dcd-1=b2c, ece-1=b2cd, fcf=cd, ede-1=c, fdf=b2d, fef=e-1 >
Subgroups: 374 in 89 conjugacy classes, 20 normal (16 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C8, C2×C4, D4, Q8, C23, C32, C12, C12, D6, C2×C6, M4(2), D8, SD16, C2×D4, C4○D4, C3×S3, C3×C6, C24, SL2(𝔽3), SL2(𝔽3), D12, C2×C12, C3×D4, C3×Q8, C22×C6, C8⋊C22, C3×C12, S3×C6, C3×M4(2), C3×D8, C3×SD16, GL2(𝔽3), C4.A4, C4.A4, C6×D4, C3×C4○D4, C3×SL2(𝔽3), C3×D12, C3×C8⋊C22, C4.3S4, C3×GL2(𝔽3), C3×C4.A4, C3×C4.3S4
Quotients: C1, C2, C3, C22, S3, C6, D6, C2×C6, C3×S3, S4, S3×C6, C2×S4, C3×S4, C4.3S4, C6×S4, C3×C4.3S4
►On 48 points
(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)
(2 4)(5 48)(6 47)(7 46)(8 45)(9 37)(10 40)(11 39)(12 38)(13 41)(14 44)(15 43)(16 42)(17 19)(21 23)(25 27)(30 32)(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), (2,4)(5,48)(6,47)(7,46)(8,45)(9,37)(10,40)(11,39)(12,38)(13,41)(14,44)(15,43)(16,42)(17,19)(21,23)(25,27)(30,32)(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), (2,4)(5,48)(6,47)(7,46)(8,45)(9,37)(10,40)(11,39)(12,38)(13,41)(14,44)(15,43)(16,42)(17,19)(21,23)(25,27)(30,32)(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)], [(2,4),(5,48),(6,47),(7,46),(8,45),(9,37),(10,40),(11,39),(12,38),(13,41),(14,44),(15,43),(16,42),(17,19),(21,23),(25,27),(30,32),(34,36)]])
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 6I | 6J | 6K | 8A | 8B | 12A | 12B | 12C | 12D | 12E | ··· | 12J | 24A | 24B | 24C | 24D |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 6 | 12 | 12 | 1 | 1 | 8 | 8 | 8 | 2 | 6 | 1 | 1 | 6 | 6 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 2 | 2 | 6 | 6 | 8 | ··· | 8 | 12 | 12 | 12 | 12 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | ||||||||
| image | C1 | C2 | C2 | C3 | C6 | C6 | S3 | D6 | C3×S3 | S3×C6 | S4 | C2×S4 | C3×S4 | C6×S4 | C4.3S4 | C3×C4.3S4 |
| kernel | C3×C4.3S4 | C3×GL2(𝔽3) | C3×C4.A4 | C4.3S4 | GL2(𝔽3) | C4.A4 | C3×C4○D4 | C3×Q8 | C4○D4 | Q8 | C12 | C6 | C4 | C2 | C3 | C1 |
| # reps | 1 | 2 | 1 | 2 | 4 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 3 | 6 |
Matrix representation of C3×C4.3S4 ►in GL4(𝔽7) generated by
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 2 | 6 | 6 | 6 |
| 0 | 5 | 2 | 5 |
| 6 | 4 | 6 | 6 |
| 6 | 3 | 4 | 1 |
| 5 | 1 | 6 | 4 |
| 0 | 2 | 2 | 3 |
| 2 | 2 | 6 | 6 |
| 1 | 4 | 4 | 1 |
| 0 | 3 | 4 | 1 |
| 2 | 6 | 0 | 4 |
| 6 | 3 | 4 | 3 |
| 4 | 5 | 5 | 4 |
| 5 | 0 | 5 | 4 |
| 6 | 0 | 5 | 1 |
| 6 | 3 | 2 | 1 |
| 4 | 5 | 4 | 5 |
| 2 | 6 | 6 | 6 |
| 4 | 3 | 0 | 1 |
| 5 | 6 | 1 | 3 |
| 1 | 3 | 4 | 1 |
G:=sub<GL(4,GF(7))| [4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[2,0,6,6,6,5,4,3,6,2,6,4,6,5,6,1],[5,0,2,1,1,2,2,4,6,2,6,4,4,3,6,1],[0,2,6,4,3,6,3,5,4,0,4,5,1,4,3,4],[5,6,6,4,0,0,3,5,5,5,2,4,4,1,1,5],[2,4,5,1,6,3,6,3,6,0,1,4,6,1,3,1] >;
C3×C4.3S4 in GAP, Magma, Sage, TeX
C_3\times C_4._3S_4
% in TeX
G:=Group("C3xC4.3S4"); // GroupNames label
G:=SmallGroup(288,904);
// by ID
G=gap.SmallGroup(288,904);
# by ID
G:=PCGroup([7,-2,-2,-3,-3,-2,2,-2,2045,1016,675,2524,655,172,1517,404,285,124]);
// 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,f*b*f=b^-1,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