direct product, non-abelian, soluble
Aliases: C3×C4.6S4, C12.20S4, GL2(𝔽3)⋊3C6, CSU2(𝔽3)⋊3C6, C4.A4⋊2C6, C2.9(C6×S4), C4.6(C3×S4), C6.46(C2×S4), Q8.4(S3×C6), (C3×Q8).22D6, (C3×GL2(𝔽3))⋊7C2, (C3×CSU2(𝔽3))⋊7C2, SL2(𝔽3).4(C2×C6), (C3×SL2(𝔽3)).16C22, (C3×C4○D4)⋊3S3, C4○D4⋊1(C3×S3), (C3×C4.A4)⋊7C2, SmallGroup(288,903)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2 — Q8 — SL2(𝔽3) — C3×SL2(𝔽3) — C3×GL2(𝔽3) — C3×C4.6S4 |
| SL2(𝔽3) — C3×C4.6S4 |
Generators and relations for C3×C4.6S4
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, bf=fb, dcd-1=b2c, ece-1=b2cd, fcf=cd, ede-1=c, fdf=b2d, fef=e-1 >
Subgroups: 286 in 83 conjugacy classes, 20 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C8, C2×C4, D4, Q8, Q8, C32, Dic3, C12, C12, D6, C2×C6, C2×C8, D8, SD16, Q16, C4○D4, C4○D4, C3×S3, C3×C6, C24, SL2(𝔽3), SL2(𝔽3), C4×S3, C2×C12, C3×D4, C3×Q8, C3×Q8, C4○D8, C3×Dic3, C3×C12, S3×C6, C2×C24, C3×D8, C3×SD16, C3×Q16, CSU2(𝔽3), GL2(𝔽3), C4.A4, C4.A4, C3×C4○D4, C3×C4○D4, C3×SL2(𝔽3), S3×C12, C3×C4○D8, C4.6S4, C3×CSU2(𝔽3), C3×GL2(𝔽3), C3×C4.A4, C3×C4.6S4
Quotients: C1, C2, C3, C22, S3, C6, D6, C2×C6, C3×S3, S4, S3×C6, C2×S4, C3×S4, C4.6S4, C6×S4, C3×C4.6S4
►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)
(1 3)(2 4)(5 48)(6 45)(7 46)(8 47)(9 39)(10 40)(11 37)(12 38)(13 43)(14 44)(15 41)(16 42)(29 31)(30 32)(33 35)(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), (1,3)(2,4)(5,48)(6,45)(7,46)(8,47)(9,39)(10,40)(11,37)(12,38)(13,43)(14,44)(15,41)(16,42)(29,31)(30,32)(33,35)(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), (1,3)(2,4)(5,48)(6,45)(7,46)(8,47)(9,39)(10,40)(11,37)(12,38)(13,43)(14,44)(15,41)(16,42)(29,31)(30,32)(33,35)(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)], [(1,3),(2,4),(5,48),(6,45),(7,46),(8,47),(9,39),(10,40),(11,37),(12,38),(13,43),(14,44),(15,41),(16,42),(29,31),(30,32),(33,35),(34,36)]])
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 6I | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | 12F | 12G | ··· | 12L | 12M | 12N | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 6 | 12 | 1 | 1 | 8 | 8 | 8 | 1 | 1 | 6 | 12 | 1 | 1 | 6 | 6 | 8 | 8 | 8 | 12 | 12 | 6 | 6 | 6 | 6 | 1 | 1 | 1 | 1 | 6 | 6 | 8 | ··· | 8 | 12 | 12 | 6 | ··· | 6 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | ||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | S3 | D6 | C3×S3 | S3×C6 | C4.6S4 | C3×C4.6S4 | S4 | C2×S4 | C3×S4 | C6×S4 | C4.6S4 | C3×C4.6S4 |
| kernel | C3×C4.6S4 | C3×CSU2(𝔽3) | C3×GL2(𝔽3) | C3×C4.A4 | C4.6S4 | CSU2(𝔽3) | GL2(𝔽3) | C4.A4 | C3×C4○D4 | C3×Q8 | C4○D4 | Q8 | C3 | C1 | C12 | C6 | C4 | C2 | C3 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 4 | 8 | 2 | 2 | 4 | 4 | 2 | 4 |
Matrix representation of C3×C4.6S4 ►in GL2(𝔽73) generated by
| 8 | 0 |
| 0 | 8 |
| 46 | 0 |
| 0 | 46 |
| 16 | 29 |
| 44 | 57 |
| 28 | 29 |
| 56 | 45 |
| 0 | 72 |
| 1 | 72 |
| 72 | 1 |
| 0 | 1 |
G:=sub<GL(2,GF(73))| [8,0,0,8],[46,0,0,46],[16,44,29,57],[28,56,29,45],[0,1,72,72],[72,0,1,1] >;
C3×C4.6S4 in GAP, Magma, Sage, TeX
C_3\times C_4._6S_4
% in TeX
G:=Group("C3xC4.6S4"); // GroupNames label
G:=SmallGroup(288,903);
// by ID
G=gap.SmallGroup(288,903);
# by ID
G:=PCGroup([7,-2,-2,-3,-3,-2,2,-2,1016,675,2524,655,172,1517,404,285,124]);
// 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,b*f=f*b,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