direct product, metabelian, supersoluble, monomial
Aliases: C3xD6:C4, D6:C12, C6.23D12, C62.19C22, (S3xC6):2C4, (C6xC12):1C2, (C2xC12):1C6, (C2xC12):1S3, C6.6(C3xD4), C2.5(S3xC12), C6.25(C4xS3), C6.4(C2xC12), C2.2(C3xD12), (C3xC6).21D4, (C2xC6).44D6, (C22xS3).C6, (C2xDic3):1C6, (C6xDic3):3C2, C22.6(S3xC6), C6.28(C3:D4), C32:5(C22:C4), (C2xC4):1(C3xS3), (S3xC2xC6).2C2, C3:1(C3xC22:C4), (C2xC6).9(C2xC6), C2.2(C3xC3:D4), (C3xC6).21(C2xC4), SmallGroup(144,79)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3xD6:C4
G = < a,b,c,d | a3=b6=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b-1, bd=db, dcd-1=b3c >
Subgroups: 168 in 76 conjugacy classes, 34 normal (30 characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C2xC4, C2xC4, C23, C32, Dic3, C12, D6, D6, C2xC6, C2xC6, C22:C4, C3xS3, C3xC6, C2xDic3, C2xC12, C2xC12, C22xS3, C22xC6, C3xDic3, C3xC12, S3xC6, S3xC6, C62, D6:C4, C3xC22:C4, C6xDic3, C6xC12, S3xC2xC6, C3xD6:C4
Quotients: C1, C2, C3, C4, C22, S3, C6, C2xC4, D4, C12, D6, C2xC6, C22:C4, C3xS3, C4xS3, D12, C3:D4, C2xC12, C3xD4, S3xC6, D6:C4, C3xC22:C4, S3xC12, C3xD12, C3xC3:D4, C3xD6:C4
►On 48 points
(1 3 5)(2 4 6)(7 9 11)(8 10 12)(13 15 17)(14 16 18)(19 21 23)(20 22 24)(25 29 27)(26 30 28)(31 35 33)(32 36 34)(37 41 39)(38 42 40)(43 47 45)(44 48 46)
(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 30)(2 29)(3 28)(4 27)(5 26)(6 25)(7 34)(8 33)(9 32)(10 31)(11 36)(12 35)(13 40)(14 39)(15 38)(16 37)(17 42)(18 41)(19 46)(20 45)(21 44)(22 43)(23 48)(24 47)
(1 23 11 17)(2 24 12 18)(3 19 7 13)(4 20 8 14)(5 21 9 15)(6 22 10 16)(25 46 31 40)(26 47 32 41)(27 48 33 42)(28 43 34 37)(29 44 35 38)(30 45 36 39)
G:=sub<Sym(48)| (1,3,5)(2,4,6)(7,9,11)(8,10,12)(13,15,17)(14,16,18)(19,21,23)(20,22,24)(25,29,27)(26,30,28)(31,35,33)(32,36,34)(37,41,39)(38,42,40)(43,47,45)(44,48,46), (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), (1,30)(2,29)(3,28)(4,27)(5,26)(6,25)(7,34)(8,33)(9,32)(10,31)(11,36)(12,35)(13,40)(14,39)(15,38)(16,37)(17,42)(18,41)(19,46)(20,45)(21,44)(22,43)(23,48)(24,47), (1,23,11,17)(2,24,12,18)(3,19,7,13)(4,20,8,14)(5,21,9,15)(6,22,10,16)(25,46,31,40)(26,47,32,41)(27,48,33,42)(28,43,34,37)(29,44,35,38)(30,45,36,39)>;
G:=Group( (1,3,5)(2,4,6)(7,9,11)(8,10,12)(13,15,17)(14,16,18)(19,21,23)(20,22,24)(25,29,27)(26,30,28)(31,35,33)(32,36,34)(37,41,39)(38,42,40)(43,47,45)(44,48,46), (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), (1,30)(2,29)(3,28)(4,27)(5,26)(6,25)(7,34)(8,33)(9,32)(10,31)(11,36)(12,35)(13,40)(14,39)(15,38)(16,37)(17,42)(18,41)(19,46)(20,45)(21,44)(22,43)(23,48)(24,47), (1,23,11,17)(2,24,12,18)(3,19,7,13)(4,20,8,14)(5,21,9,15)(6,22,10,16)(25,46,31,40)(26,47,32,41)(27,48,33,42)(28,43,34,37)(29,44,35,38)(30,45,36,39) );
G=PermutationGroup([[(1,3,5),(2,4,6),(7,9,11),(8,10,12),(13,15,17),(14,16,18),(19,21,23),(20,22,24),(25,29,27),(26,30,28),(31,35,33),(32,36,34),(37,41,39),(38,42,40),(43,47,45),(44,48,46)], [(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)], [(1,30),(2,29),(3,28),(4,27),(5,26),(6,25),(7,34),(8,33),(9,32),(10,31),(11,36),(12,35),(13,40),(14,39),(15,38),(16,37),(17,42),(18,41),(19,46),(20,45),(21,44),(22,43),(23,48),(24,47)], [(1,23,11,17),(2,24,12,18),(3,19,7,13),(4,20,8,14),(5,21,9,15),(6,22,10,16),(25,46,31,40),(26,47,32,41),(27,48,33,42),(28,43,34,37),(29,44,35,38),(30,45,36,39)]])
C3xD6:C4 is a maximal subgroup of
Dic3.D12 C62.23C23 C62.24C23 C62.28C23 C62.29C23 C62.31C23 C62.32C23 C62.47C23 C62.48C23 C62.49C23 Dic3:4D12 C62.51C23 C62.55C23 D6:1Dic6 D6.D12 D6.9D12 D6:2Dic6 D6:3Dic6 D6:4Dic6 C62.72C23 C62.75C23 D6:D12 C62.77C23 Dic3:3D12 C62.82C23 C62.83C23 C62.85C23 C62.91C23 D6:4D12 D6:5D12 C12xD12 C3xS3xC22:C4 C12xC3:D4 C62.21D6 D18:C12 C62.31D6
C3xD6:C4 is a maximal quotient of
C62.21D6 D18:C12
54 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 4C | 4D | 6A | ··· | 6F | 6G | ··· | 6O | 6P | 6Q | 6R | 6S | 12A | ··· | 12P | 12Q | 12R | 12S | 12T |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 12 | ··· | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 6 | 6 | 1 | ··· | 1 | 2 | ··· | 2 | 6 | 6 | 6 | 6 | 2 | ··· | 2 | 6 | 6 | 6 | 6 |
54 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | ||||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C4 | C6 | C6 | C6 | C12 | S3 | D4 | D6 | C3xS3 | C4xS3 | D12 | C3:D4 | C3xD4 | S3xC6 | S3xC12 | C3xD12 | C3xC3:D4 |
| kernel | C3xD6:C4 | C6xDic3 | C6xC12 | S3xC2xC6 | D6:C4 | S3xC6 | C2xDic3 | C2xC12 | C22xS3 | D6 | C2xC12 | C3xC6 | C2xC6 | C2xC4 | C6 | C6 | C6 | C6 | C22 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 4 | 2 | 2 | 2 | 8 | 1 | 2 | 1 | 2 | 2 | 2 | 2 | 4 | 2 | 4 | 4 | 4 |
Matrix representation of C3xD6:C4 ►in GL3(F13) generated by
| 9 | 0 | 0 |
| 0 | 3 | 0 |
| 0 | 0 | 3 |
| 1 | 0 | 0 |
| 0 | 4 | 0 |
| 0 | 0 | 10 |
| 12 | 0 | 0 |
| 0 | 0 | 10 |
| 0 | 4 | 0 |
| 5 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 12 |
G:=sub<GL(3,GF(13))| [9,0,0,0,3,0,0,0,3],[1,0,0,0,4,0,0,0,10],[12,0,0,0,0,4,0,10,0],[5,0,0,0,1,0,0,0,12] >;
C3xD6:C4 in GAP, Magma, Sage, TeX
C_3\times D_6\rtimes C_4
% in TeX
G:=Group("C3xD6:C4"); // GroupNames label
G:=SmallGroup(144,79);
// by ID
G=gap.SmallGroup(144,79);
# by ID
G:=PCGroup([6,-2,-2,-3,-2,-2,-3,313,79,3461]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^6=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,b*d=d*b,d*c*d^-1=b^3*c>;
// generators/relations