direct product, metabelian, supersoluble, monomial
Aliases: C3×D6⋊C4, D6⋊C12, C6.23D12, C62.19C22, (S3×C6)⋊2C4, (C6×C12)⋊1C2, (C2×C12)⋊1C6, (C2×C12)⋊1S3, C6.6(C3×D4), C2.5(S3×C12), C6.25(C4×S3), C6.4(C2×C12), C2.2(C3×D12), (C3×C6).21D4, (C2×C6).44D6, (C22×S3).C6, (C2×Dic3)⋊1C6, (C6×Dic3)⋊3C2, C22.6(S3×C6), C6.28(C3⋊D4), C32⋊5(C22⋊C4), (C2×C4)⋊1(C3×S3), (S3×C2×C6).2C2, C3⋊1(C3×C22⋊C4), (C2×C6).9(C2×C6), C2.2(C3×C3⋊D4), (C3×C6).21(C2×C4), SmallGroup(144,79)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×D6⋊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, C2×C4, C2×C4, C23, C32, Dic3, C12, D6, D6, C2×C6, C2×C6, C22⋊C4, C3×S3, C3×C6, C2×Dic3, C2×C12, C2×C12, C22×S3, C22×C6, C3×Dic3, C3×C12, S3×C6, S3×C6, C62, D6⋊C4, C3×C22⋊C4, C6×Dic3, C6×C12, S3×C2×C6, C3×D6⋊C4
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, D4, C12, D6, C2×C6, C22⋊C4, C3×S3, C4×S3, D12, C3⋊D4, C2×C12, C3×D4, S3×C6, D6⋊C4, C3×C22⋊C4, S3×C12, C3×D12, C3×C3⋊D4, C3×D6⋊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)]])
C3×D6⋊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 C12×D12 C3×S3×C22⋊C4 C12×C3⋊D4 C62.21D6 D18⋊C12 C62.31D6
C3×D6⋊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 | C3×S3 | C4×S3 | D12 | C3⋊D4 | C3×D4 | S3×C6 | S3×C12 | C3×D12 | C3×C3⋊D4 |
| kernel | C3×D6⋊C4 | C6×Dic3 | C6×C12 | S3×C2×C6 | D6⋊C4 | S3×C6 | C2×Dic3 | C2×C12 | C22×S3 | D6 | C2×C12 | C3×C6 | C2×C6 | C2×C4 | 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 C3×D6⋊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] >;
C3×D6⋊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