direct product, non-abelian, soluble
Aliases: C3×D4.A4, Q8.3C62, 2- 1+4⋊C32, D4.(C3×A4), C4.A4⋊4C6, C4.3(C6×A4), (C6×Q8)⋊3C6, (C3×D4).3A4, C12.13(C2×A4), C22.5(C6×A4), C6.26(C22×A4), (C3×2- 1+4)⋊C3, (C2×SL2(𝔽3))⋊1C6, (C6×SL2(𝔽3))⋊8C2, SL2(𝔽3).5(C2×C6), (C3×SL2(𝔽3)).17C22, C4○D4⋊(C3×C6), (C2×Q8)⋊(C3×C6), C2.7(A4×C2×C6), (C3×C4○D4)⋊3C6, (C3×C4.A4)⋊9C2, (C2×C6).16(C2×A4), (C3×Q8).17(C2×C6), SmallGroup(288,985)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2 — Q8 — C3×Q8 — C3×SL2(𝔽3) — C6×SL2(𝔽3) — C3×D4.A4 |
| Q8 — C3×D4.A4 |
Generators and relations for C3×D4.A4
G = < a,b,c,d,e,f | a3=b4=c2=f3=1, d2=e2=b2, 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=b2d, fdf-1=b2de, fef-1=d >
Subgroups: 336 in 118 conjugacy classes, 42 normal (17 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, C6, C6, C2×C4, D4, D4, Q8, Q8, C32, C12, C12, C2×C6, C2×C6, C2×Q8, C2×Q8, C4○D4, C4○D4, C3×C6, SL2(𝔽3), C2×C12, C3×D4, C3×D4, C3×Q8, C3×Q8, 2- 1+4, C3×C12, C62, C2×SL2(𝔽3), C4.A4, C6×Q8, C6×Q8, C3×C4○D4, C3×C4○D4, C3×SL2(𝔽3), D4×C32, D4.A4, C3×2- 1+4, C6×SL2(𝔽3), C3×C4.A4, C3×D4.A4
Quotients: C1, C2, C3, C22, C6, C32, A4, C2×C6, C3×C6, C2×A4, C3×A4, C62, C22×A4, C6×A4, D4.A4, A4×C2×C6, C3×D4.A4
►On 48 points
(1 14 10)(2 15 11)(3 16 12)(4 13 9)(5 34 30)(6 35 31)(7 36 32)(8 33 29)(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)
(2 4)(5 7)(9 11)(13 15)(17 19)(21 23)(25 27)(30 32)(34 36)(38 40)(42 44)(46 48)
(1 18 3 20)(2 19 4 17)(5 46 7 48)(6 47 8 45)(9 21 11 23)(10 22 12 24)(13 27 15 25)(14 28 16 26)(29 37 31 39)(30 38 32 40)(33 41 35 43)(34 42 36 44)
(1 39 3 37)(2 40 4 38)(5 25 7 27)(6 26 8 28)(9 42 11 44)(10 43 12 41)(13 46 15 48)(14 47 16 45)(17 30 19 32)(18 31 20 29)(21 34 23 36)(22 35 24 33)
(1 14 10)(2 15 11)(3 16 12)(4 13 9)(5 23 40)(6 24 37)(7 21 38)(8 22 39)(17 46 36)(18 47 33)(19 48 34)(20 45 35)(25 44 30)(26 41 31)(27 42 32)(28 43 29)
G:=sub<Sym(48)| (1,14,10)(2,15,11)(3,16,12)(4,13,9)(5,34,30)(6,35,31)(7,36,32)(8,33,29)(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), (2,4)(5,7)(9,11)(13,15)(17,19)(21,23)(25,27)(30,32)(34,36)(38,40)(42,44)(46,48), (1,18,3,20)(2,19,4,17)(5,46,7,48)(6,47,8,45)(9,21,11,23)(10,22,12,24)(13,27,15,25)(14,28,16,26)(29,37,31,39)(30,38,32,40)(33,41,35,43)(34,42,36,44), (1,39,3,37)(2,40,4,38)(5,25,7,27)(6,26,8,28)(9,42,11,44)(10,43,12,41)(13,46,15,48)(14,47,16,45)(17,30,19,32)(18,31,20,29)(21,34,23,36)(22,35,24,33), (1,14,10)(2,15,11)(3,16,12)(4,13,9)(5,23,40)(6,24,37)(7,21,38)(8,22,39)(17,46,36)(18,47,33)(19,48,34)(20,45,35)(25,44,30)(26,41,31)(27,42,32)(28,43,29)>;
G:=Group( (1,14,10)(2,15,11)(3,16,12)(4,13,9)(5,34,30)(6,35,31)(7,36,32)(8,33,29)(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), (2,4)(5,7)(9,11)(13,15)(17,19)(21,23)(25,27)(30,32)(34,36)(38,40)(42,44)(46,48), (1,18,3,20)(2,19,4,17)(5,46,7,48)(6,47,8,45)(9,21,11,23)(10,22,12,24)(13,27,15,25)(14,28,16,26)(29,37,31,39)(30,38,32,40)(33,41,35,43)(34,42,36,44), (1,39,3,37)(2,40,4,38)(5,25,7,27)(6,26,8,28)(9,42,11,44)(10,43,12,41)(13,46,15,48)(14,47,16,45)(17,30,19,32)(18,31,20,29)(21,34,23,36)(22,35,24,33), (1,14,10)(2,15,11)(3,16,12)(4,13,9)(5,23,40)(6,24,37)(7,21,38)(8,22,39)(17,46,36)(18,47,33)(19,48,34)(20,45,35)(25,44,30)(26,41,31)(27,42,32)(28,43,29) );
G=PermutationGroup([[(1,14,10),(2,15,11),(3,16,12),(4,13,9),(5,34,30),(6,35,31),(7,36,32),(8,33,29),(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)], [(2,4),(5,7),(9,11),(13,15),(17,19),(21,23),(25,27),(30,32),(34,36),(38,40),(42,44),(46,48)], [(1,18,3,20),(2,19,4,17),(5,46,7,48),(6,47,8,45),(9,21,11,23),(10,22,12,24),(13,27,15,25),(14,28,16,26),(29,37,31,39),(30,38,32,40),(33,41,35,43),(34,42,36,44)], [(1,39,3,37),(2,40,4,38),(5,25,7,27),(6,26,8,28),(9,42,11,44),(10,43,12,41),(13,46,15,48),(14,47,16,45),(17,30,19,32),(18,31,20,29),(21,34,23,36),(22,35,24,33)], [(1,14,10),(2,15,11),(3,16,12),(4,13,9),(5,23,40),(6,24,37),(7,21,38),(8,22,39),(17,46,36),(18,47,33),(19,48,34),(20,45,35),(25,44,30),(26,41,31),(27,42,32),(28,43,29)]])
57 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3A | 3B | 3C | ··· | 3H | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 6D | 6E | 6F | 6G | ··· | 6L | 6M | 6N | 6O | ··· | 6Z | 12A | 12B | 12C | ··· | 12H | 12I | ··· | 12N |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | ··· | 3 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | ··· | 6 | 6 | 6 | 6 | ··· | 6 | 12 | 12 | 12 | ··· | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 2 | 2 | 6 | 1 | 1 | 4 | ··· | 4 | 2 | 6 | 6 | 6 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 6 | 6 | 8 | ··· | 8 | 2 | 2 | 6 | ··· | 6 | 8 | ··· | 8 |
57 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | - | |||||||||||
| image | C1 | C2 | C2 | C3 | C3 | C6 | C6 | C6 | C6 | A4 | C2×A4 | C2×A4 | C3×A4 | C6×A4 | C6×A4 | D4.A4 | D4.A4 | C3×D4.A4 |
| kernel | C3×D4.A4 | C6×SL2(𝔽3) | C3×C4.A4 | D4.A4 | C3×2- 1+4 | C2×SL2(𝔽3) | C4.A4 | C6×Q8 | C3×C4○D4 | C3×D4 | C12 | C2×C6 | D4 | C4 | C22 | C3 | C3 | C1 |
| # reps | 1 | 2 | 1 | 6 | 2 | 12 | 6 | 4 | 2 | 1 | 1 | 2 | 2 | 2 | 4 | 1 | 2 | 6 |
Matrix representation of C3×D4.A4 ►in GL4(𝔽7) generated by
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 3 | 5 | 6 | 1 |
| 5 | 5 | 3 | 2 |
| 6 | 3 | 0 | 5 |
| 6 | 5 | 2 | 6 |
| 3 | 5 | 1 | 4 |
| 5 | 5 | 2 | 3 |
| 6 | 3 | 0 | 5 |
| 6 | 5 | 2 | 6 |
| 0 | 5 | 0 | 5 |
| 4 | 0 | 2 | 4 |
| 0 | 0 | 4 | 5 |
| 0 | 0 | 5 | 3 |
| 2 | 1 | 2 | 0 |
| 2 | 5 | 2 | 2 |
| 0 | 0 | 4 | 6 |
| 0 | 0 | 3 | 3 |
| 1 | 4 | 1 | 5 |
| 0 | 4 | 3 | 5 |
| 0 | 0 | 2 | 6 |
| 0 | 0 | 5 | 3 |
G:=sub<GL(4,GF(7))| [4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[3,5,6,6,5,5,3,5,6,3,0,2,1,2,5,6],[3,5,6,6,5,5,3,5,1,2,0,2,4,3,5,6],[0,4,0,0,5,0,0,0,0,2,4,5,5,4,5,3],[2,2,0,0,1,5,0,0,2,2,4,3,0,2,6,3],[1,0,0,0,4,4,0,0,1,3,2,5,5,5,6,3] >;
C3×D4.A4 in GAP, Magma, Sage, TeX
C_3\times D_4.A_4
% in TeX
G:=Group("C3xD4.A4"); // GroupNames label
G:=SmallGroup(288,985);
// by ID
G=gap.SmallGroup(288,985);
# by ID
G:=PCGroup([7,-2,-2,-3,-3,-2,2,-2,2045,648,172,1153,285,124]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^3=b^4=c^2=f^3=1,d^2=e^2=b^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=b^2*d,f*d*f^-1=b^2*d*e,f*e*f^-1=d>;
// generators/relations