direct product, metabelian, supersoluble, monomial
Aliases: C3×D4.D6, C24.42D6, Dic12⋊6C6, C8.2(S3×C6), (S3×Q8)⋊5C6, C8⋊S3⋊2C6, C24.9(C2×C6), D4⋊2S3.C6, D4.S3⋊4C6, C3⋊Q16⋊1C6, D4.4(S3×C6), D6.8(C3×D4), C6.32(C6×D4), SD16⋊2(C3×S3), (C3×SD16)⋊6S3, (C3×SD16)⋊2C6, (S3×C6).44D4, (C3×D4).27D6, C6.192(S3×D4), (C3×Q8).48D6, Q8.11(S3×C6), C12.6(C22×C6), Dic6.2(C2×C6), (C3×Dic12)⋊14C2, (C3×C24).32C22, (C3×C12).77C23, Dic3.10(C3×D4), (C3×Dic3).47D4, (C32×SD16)⋊2C2, (S3×C12).28C22, C12.157(C22×S3), C32⋊18(C8.C22), (C3×Dic6).25C22, (D4×C32).14C22, (Q8×C32).11C22, C4.6(S3×C2×C6), (C3×S3×Q8)⋊5C2, C3⋊C8.1(C2×C6), C2.20(C3×S3×D4), (C3×C8⋊S3)⋊6C2, (C4×S3).3(C2×C6), (C3×D4).4(C2×C6), C3⋊2(C3×C8.C22), (C3×Q8).6(C2×C6), (C3×D4.S3)⋊10C2, (C3×C3⋊Q16)⋊11C2, (C3×C6).220(C2×D4), (C3×C3⋊C8).20C22, (C3×D4⋊2S3).2C2, SmallGroup(288,686)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×D4.D6
G = < a,b,c,d,e | a3=b4=c2=1, d6=e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=dbd-1=ebe-1=b-1, dcd-1=bc, ece-1=b-1c, ede-1=d5 >
Subgroups: 306 in 130 conjugacy classes, 54 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C8, C8, C2×C4, D4, D4, Q8, Q8, C32, Dic3, Dic3, C12, C12, D6, C2×C6, M4(2), SD16, SD16, Q16, C2×Q8, C4○D4, C3×S3, C3×C6, C3×C6, C3⋊C8, C24, C24, Dic6, Dic6, C4×S3, C4×S3, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C3×D4, C3×Q8, C3×Q8, C8.C22, C3×Dic3, C3×Dic3, C3×C12, C3×C12, S3×C6, C62, C8⋊S3, Dic12, D4.S3, C3⋊Q16, C3×M4(2), C3×SD16, C3×SD16, C3×Q16, D4⋊2S3, S3×Q8, C6×Q8, C3×C4○D4, C3×C3⋊C8, C3×C24, C3×Dic6, C3×Dic6, S3×C12, S3×C12, C6×Dic3, C3×C3⋊D4, D4×C32, Q8×C32, D4.D6, C3×C8.C22, C3×C8⋊S3, C3×Dic12, C3×D4.S3, C3×C3⋊Q16, C32×SD16, C3×D4⋊2S3, C3×S3×Q8, C3×D4.D6
Quotients: C1, C2, C3, C22, S3, C6, D4, C23, D6, C2×C6, C2×D4, C3×S3, C3×D4, C22×S3, C22×C6, C8.C22, S3×C6, S3×D4, C6×D4, S3×C2×C6, D4.D6, C3×C8.C22, C3×S3×D4, C3×D4.D6
►On 48 points
(1 9 5)(2 10 6)(3 11 7)(4 12 8)(13 17 21)(14 18 22)(15 19 23)(16 20 24)(25 29 33)(26 30 34)(27 31 35)(28 32 36)(37 45 41)(38 46 42)(39 47 43)(40 48 44)
(1 40 7 46)(2 47 8 41)(3 42 9 48)(4 37 10 43)(5 44 11 38)(6 39 12 45)(13 28 19 34)(14 35 20 29)(15 30 21 36)(16 25 22 31)(17 32 23 26)(18 27 24 33)
(1 46)(2 8)(3 48)(4 10)(5 38)(6 12)(7 40)(9 42)(11 44)(14 35)(16 25)(18 27)(20 29)(22 31)(24 33)(26 32)(28 34)(30 36)
(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 15 7 21)(2 20 8 14)(3 13 9 19)(4 18 10 24)(5 23 11 17)(6 16 12 22)(25 39 31 45)(26 44 32 38)(27 37 33 43)(28 42 34 48)(29 47 35 41)(30 40 36 46)
G:=sub<Sym(48)| (1,9,5)(2,10,6)(3,11,7)(4,12,8)(13,17,21)(14,18,22)(15,19,23)(16,20,24)(25,29,33)(26,30,34)(27,31,35)(28,32,36)(37,45,41)(38,46,42)(39,47,43)(40,48,44), (1,40,7,46)(2,47,8,41)(3,42,9,48)(4,37,10,43)(5,44,11,38)(6,39,12,45)(13,28,19,34)(14,35,20,29)(15,30,21,36)(16,25,22,31)(17,32,23,26)(18,27,24,33), (1,46)(2,8)(3,48)(4,10)(5,38)(6,12)(7,40)(9,42)(11,44)(14,35)(16,25)(18,27)(20,29)(22,31)(24,33)(26,32)(28,34)(30,36), (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), (1,15,7,21)(2,20,8,14)(3,13,9,19)(4,18,10,24)(5,23,11,17)(6,16,12,22)(25,39,31,45)(26,44,32,38)(27,37,33,43)(28,42,34,48)(29,47,35,41)(30,40,36,46)>;
G:=Group( (1,9,5)(2,10,6)(3,11,7)(4,12,8)(13,17,21)(14,18,22)(15,19,23)(16,20,24)(25,29,33)(26,30,34)(27,31,35)(28,32,36)(37,45,41)(38,46,42)(39,47,43)(40,48,44), (1,40,7,46)(2,47,8,41)(3,42,9,48)(4,37,10,43)(5,44,11,38)(6,39,12,45)(13,28,19,34)(14,35,20,29)(15,30,21,36)(16,25,22,31)(17,32,23,26)(18,27,24,33), (1,46)(2,8)(3,48)(4,10)(5,38)(6,12)(7,40)(9,42)(11,44)(14,35)(16,25)(18,27)(20,29)(22,31)(24,33)(26,32)(28,34)(30,36), (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), (1,15,7,21)(2,20,8,14)(3,13,9,19)(4,18,10,24)(5,23,11,17)(6,16,12,22)(25,39,31,45)(26,44,32,38)(27,37,33,43)(28,42,34,48)(29,47,35,41)(30,40,36,46) );
G=PermutationGroup([[(1,9,5),(2,10,6),(3,11,7),(4,12,8),(13,17,21),(14,18,22),(15,19,23),(16,20,24),(25,29,33),(26,30,34),(27,31,35),(28,32,36),(37,45,41),(38,46,42),(39,47,43),(40,48,44)], [(1,40,7,46),(2,47,8,41),(3,42,9,48),(4,37,10,43),(5,44,11,38),(6,39,12,45),(13,28,19,34),(14,35,20,29),(15,30,21,36),(16,25,22,31),(17,32,23,26),(18,27,24,33)], [(1,46),(2,8),(3,48),(4,10),(5,38),(6,12),(7,40),(9,42),(11,44),(14,35),(16,25),(18,27),(20,29),(22,31),(24,33),(26,32),(28,34),(30,36)], [(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)], [(1,15,7,21),(2,20,8,14),(3,13,9,19),(4,18,10,24),(5,23,11,17),(6,16,12,22),(25,39,31,45),(26,44,32,38),(27,37,33,43),(28,42,34,48),(29,47,35,41),(30,40,36,46)]])
54 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 6I | 6J | 6K | 6L | 8A | 8B | 12A | 12B | 12C | ··· | 12G | 12H | 12I | 12J | 12K | 12L | 12M | 12N | 12O | 12P | 24A | ··· | 24H | 24I | 24J |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 12 | 12 | 12 | ··· | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | ··· | 24 | 24 | 24 |
| size | 1 | 1 | 4 | 6 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 6 | 12 | 12 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 6 | 6 | 8 | 8 | 8 | 4 | 12 | 2 | 2 | 4 | ··· | 4 | 6 | 6 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 4 | ··· | 4 | 12 | 12 |
54 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - | |||||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | C6 | C6 | C6 | S3 | D4 | D4 | D6 | D6 | D6 | C3×S3 | C3×D4 | C3×D4 | S3×C6 | S3×C6 | S3×C6 | C8.C22 | S3×D4 | D4.D6 | C3×C8.C22 | C3×S3×D4 | C3×D4.D6 |
| kernel | C3×D4.D6 | C3×C8⋊S3 | C3×Dic12 | C3×D4.S3 | C3×C3⋊Q16 | C32×SD16 | C3×D4⋊2S3 | C3×S3×Q8 | D4.D6 | C8⋊S3 | Dic12 | D4.S3 | C3⋊Q16 | C3×SD16 | D4⋊2S3 | S3×Q8 | C3×SD16 | C3×Dic3 | S3×C6 | C24 | C3×D4 | C3×Q8 | SD16 | Dic3 | D6 | C8 | D4 | Q8 | C32 | C6 | C3 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 4 |
Matrix representation of C3×D4.D6 ►in GL4(𝔽73) generated by
| 8 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 |
| 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 8 |
| 0 | 1 | 0 | 0 |
| 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 72 | 0 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 |
| 0 | 0 | 72 | 0 |
| 19 | 54 | 0 | 0 |
| 54 | 54 | 0 | 0 |
| 0 | 0 | 48 | 25 |
| 0 | 0 | 25 | 25 |
| 0 | 0 | 48 | 25 |
| 0 | 0 | 25 | 25 |
| 19 | 54 | 0 | 0 |
| 54 | 54 | 0 | 0 |
G:=sub<GL(4,GF(73))| [8,0,0,0,0,8,0,0,0,0,8,0,0,0,0,8],[0,72,0,0,1,0,0,0,0,0,0,72,0,0,1,0],[0,1,0,0,1,0,0,0,0,0,0,72,0,0,72,0],[19,54,0,0,54,54,0,0,0,0,48,25,0,0,25,25],[0,0,19,54,0,0,54,54,48,25,0,0,25,25,0,0] >;
C3×D4.D6 in GAP, Magma, Sage, TeX
C_3\times D_4.D_6
% in TeX
G:=Group("C3xD4.D6"); // GroupNames label
G:=SmallGroup(288,686);
// by ID
G=gap.SmallGroup(288,686);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,344,1094,303,1271,648,102,9414]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^4=c^2=1,d^6=e^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=d*b*d^-1=e*b*e^-1=b^-1,d*c*d^-1=b*c,e*c*e^-1=b^-1*c,e*d*e^-1=d^5>;
// generators/relations