direct product, non-abelian, soluble, monomial
Aliases: C3×C6×S4, A4⋊C62, C62⋊11D6, (C6×A4)⋊3C6, (C2×C62)⋊1S3, C23⋊(S3×C32), (C32×A4)⋊6C22, C22⋊(S3×C3×C6), (C2×A4)⋊(C3×C6), (A4×C3×C6)⋊1C2, (C2×C6)⋊3(S3×C6), (C3×A4)⋊4(C2×C6), (C22×C6)⋊1(C3×S3), SmallGroup(432,760)
Series: Derived ►Chief ►Lower central ►Upper central
| A4 — C3×C6×S4 |
Generators and relations for C3×C6×S4
G = < a,b,c,d,e,f | a3=b6=c2=d2=e3=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, ece-1=fcf=cd=dc, ede-1=c, df=fd, fef=e-1 >
Subgroups: 748 in 230 conjugacy classes, 54 normal (14 characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C2×C4, D4, C23, C23, C32, C32, C12, A4, A4, D6, C2×C6, C2×C6, C2×D4, C3×S3, C3×C6, C3×C6, C2×C12, C3×D4, S4, C2×A4, C2×A4, C22×C6, C22×C6, C33, C3×C12, C3×A4, C3×A4, S3×C6, C62, C62, C6×D4, C2×S4, S3×C32, C32×C6, C6×C12, D4×C32, C3×S4, C6×A4, C6×A4, C2×C62, C2×C62, C32×A4, S3×C3×C6, D4×C3×C6, C6×S4, C32×S4, A4×C3×C6, C3×C6×S4
Quotients: C1, C2, C3, C22, S3, C6, C32, D6, C2×C6, C3×S3, C3×C6, S4, S3×C6, C62, C2×S4, S3×C32, C3×S4, S3×C3×C6, C6×S4, C32×S4, C3×C6×S4
►On 54 points
Generators in S54
(1 49 33)(2 50 34)(3 51 35)(4 52 36)(5 53 31)(6 54 32)(7 42 20)(8 37 21)(9 38 22)(10 39 23)(11 40 24)(12 41 19)(13 44 28)(14 45 29)(15 46 30)(16 47 25)(17 48 26)(18 43 27)
(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)(49 50 51 52 53 54)
(1 4)(2 5)(3 6)(13 16)(14 17)(15 18)(25 28)(26 29)(27 30)(31 34)(32 35)(33 36)(43 46)(44 47)(45 48)(49 52)(50 53)(51 54)
(7 10)(8 11)(9 12)(13 16)(14 17)(15 18)(19 22)(20 23)(21 24)(25 28)(26 29)(27 30)(37 40)(38 41)(39 42)(43 46)(44 47)(45 48)
(1 29 24)(2 30 19)(3 25 20)(4 26 21)(5 27 22)(6 28 23)(7 51 16)(8 52 17)(9 53 18)(10 54 13)(11 49 14)(12 50 15)(31 43 38)(32 44 39)(33 45 40)(34 46 41)(35 47 42)(36 48 37)
(1 4)(2 5)(3 6)(7 13)(8 14)(9 15)(10 16)(11 17)(12 18)(19 27)(20 28)(21 29)(22 30)(23 25)(24 26)(31 34)(32 35)(33 36)(37 45)(38 46)(39 47)(40 48)(41 43)(42 44)(49 52)(50 53)(51 54)
G:=sub<Sym(54)| (1,49,33)(2,50,34)(3,51,35)(4,52,36)(5,53,31)(6,54,32)(7,42,20)(8,37,21)(9,38,22)(10,39,23)(11,40,24)(12,41,19)(13,44,28)(14,45,29)(15,46,30)(16,47,25)(17,48,26)(18,43,27), (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)(49,50,51,52,53,54), (1,4)(2,5)(3,6)(13,16)(14,17)(15,18)(25,28)(26,29)(27,30)(31,34)(32,35)(33,36)(43,46)(44,47)(45,48)(49,52)(50,53)(51,54), (7,10)(8,11)(9,12)(13,16)(14,17)(15,18)(19,22)(20,23)(21,24)(25,28)(26,29)(27,30)(37,40)(38,41)(39,42)(43,46)(44,47)(45,48), (1,29,24)(2,30,19)(3,25,20)(4,26,21)(5,27,22)(6,28,23)(7,51,16)(8,52,17)(9,53,18)(10,54,13)(11,49,14)(12,50,15)(31,43,38)(32,44,39)(33,45,40)(34,46,41)(35,47,42)(36,48,37), (1,4)(2,5)(3,6)(7,13)(8,14)(9,15)(10,16)(11,17)(12,18)(19,27)(20,28)(21,29)(22,30)(23,25)(24,26)(31,34)(32,35)(33,36)(37,45)(38,46)(39,47)(40,48)(41,43)(42,44)(49,52)(50,53)(51,54)>;
G:=Group( (1,49,33)(2,50,34)(3,51,35)(4,52,36)(5,53,31)(6,54,32)(7,42,20)(8,37,21)(9,38,22)(10,39,23)(11,40,24)(12,41,19)(13,44,28)(14,45,29)(15,46,30)(16,47,25)(17,48,26)(18,43,27), (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)(49,50,51,52,53,54), (1,4)(2,5)(3,6)(13,16)(14,17)(15,18)(25,28)(26,29)(27,30)(31,34)(32,35)(33,36)(43,46)(44,47)(45,48)(49,52)(50,53)(51,54), (7,10)(8,11)(9,12)(13,16)(14,17)(15,18)(19,22)(20,23)(21,24)(25,28)(26,29)(27,30)(37,40)(38,41)(39,42)(43,46)(44,47)(45,48), (1,29,24)(2,30,19)(3,25,20)(4,26,21)(5,27,22)(6,28,23)(7,51,16)(8,52,17)(9,53,18)(10,54,13)(11,49,14)(12,50,15)(31,43,38)(32,44,39)(33,45,40)(34,46,41)(35,47,42)(36,48,37), (1,4)(2,5)(3,6)(7,13)(8,14)(9,15)(10,16)(11,17)(12,18)(19,27)(20,28)(21,29)(22,30)(23,25)(24,26)(31,34)(32,35)(33,36)(37,45)(38,46)(39,47)(40,48)(41,43)(42,44)(49,52)(50,53)(51,54) );
G=PermutationGroup([[(1,49,33),(2,50,34),(3,51,35),(4,52,36),(5,53,31),(6,54,32),(7,42,20),(8,37,21),(9,38,22),(10,39,23),(11,40,24),(12,41,19),(13,44,28),(14,45,29),(15,46,30),(16,47,25),(17,48,26),(18,43,27)], [(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),(49,50,51,52,53,54)], [(1,4),(2,5),(3,6),(13,16),(14,17),(15,18),(25,28),(26,29),(27,30),(31,34),(32,35),(33,36),(43,46),(44,47),(45,48),(49,52),(50,53),(51,54)], [(7,10),(8,11),(9,12),(13,16),(14,17),(15,18),(19,22),(20,23),(21,24),(25,28),(26,29),(27,30),(37,40),(38,41),(39,42),(43,46),(44,47),(45,48)], [(1,29,24),(2,30,19),(3,25,20),(4,26,21),(5,27,22),(6,28,23),(7,51,16),(8,52,17),(9,53,18),(10,54,13),(11,49,14),(12,50,15),(31,43,38),(32,44,39),(33,45,40),(34,46,41),(35,47,42),(36,48,37)], [(1,4),(2,5),(3,6),(7,13),(8,14),(9,15),(10,16),(11,17),(12,18),(19,27),(20,28),(21,29),(22,30),(23,25),(24,26),(31,34),(32,35),(33,36),(37,45),(38,46),(39,47),(40,48),(41,43),(42,44),(49,52),(50,53),(51,54)]])
90 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | ··· | 3H | 3I | ··· | 3Q | 4A | 4B | 6A | ··· | 6H | 6I | ··· | 6X | 6Y | ··· | 6AN | 6AO | ··· | 6AW | 12A | ··· | 12P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | ··· | 3 | 3 | ··· | 3 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | 3 | 3 | 6 | 6 | 1 | ··· | 1 | 8 | ··· | 8 | 6 | 6 | 1 | ··· | 1 | 3 | ··· | 3 | 6 | ··· | 6 | 8 | ··· | 8 | 6 | ··· | 6 |
90 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 |
| type | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C3 | C6 | C6 | S3 | D6 | C3×S3 | S3×C6 | S4 | C2×S4 | C3×S4 | C6×S4 |
| kernel | C3×C6×S4 | C32×S4 | A4×C3×C6 | C6×S4 | C3×S4 | C6×A4 | C2×C62 | C62 | C22×C6 | C2×C6 | C3×C6 | C32 | C6 | C3 |
| # reps | 1 | 2 | 1 | 8 | 16 | 8 | 1 | 1 | 8 | 8 | 2 | 2 | 16 | 16 |
Matrix representation of C3×C6×S4 ►in GL5(𝔽13)
| 3 | 0 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 |
| 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 9 |
| 4 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 12 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 12 | 0 |
| 0 | 0 | 1 | 0 | 12 |
| 12 | 12 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 11 |
| 0 | 0 | 0 | 0 | 12 |
| 0 | 0 | 0 | 1 | 12 |
| 12 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 |
| 0 | 0 | 0 | 12 | 0 |
G:=sub<GL(5,GF(13))| [3,0,0,0,0,0,3,0,0,0,0,0,9,0,0,0,0,0,9,0,0,0,0,0,9],[4,0,0,0,0,0,4,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,1,0,0,0,0,0,12,0,12,0,0,0,12,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,1,1,0,0,0,12,0,0,0,0,0,12],[12,1,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,11,12,12],[12,1,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,12,0] >;
C3×C6×S4 in GAP, Magma, Sage, TeX
C_3\times C_6\times S_4
% in TeX
G:=Group("C3xC6xS4"); // GroupNames label
G:=SmallGroup(432,760);
// by ID
G=gap.SmallGroup(432,760);
# by ID
G:=PCGroup([7,-2,-2,-3,-3,-3,-2,2,2524,9077,285,5298,475]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^3=b^6=c^2=d^2=e^3=f^2=1,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,b*f=f*b,e*c*e^-1=f*c*f=c*d=d*c,e*d*e^-1=c,d*f=f*d,f*e*f=e^-1>;
// generators/relations