direct product, metabelian, supersoluble, monomial
Aliases: C3×Dic3.D4, C62⋊5Q8, C62.169C23, C6.4(C6×Q8), C4⋊Dic3⋊2C6, (C2×C6)⋊6Dic6, C6.16(C6×D4), Dic3⋊C4⋊4C6, (C2×Dic6)⋊2C6, C6.174(S3×D4), C2.6(C6×Dic6), (C6×Dic6)⋊26C2, (C2×C12).230D6, C23.22(S3×C6), Dic3.6(C3×D4), C6.50(C2×Dic6), C22⋊3(C3×Dic6), (C3×Dic3).43D4, C6.D4.2C6, (C22×C6).103D6, C32⋊16(C22⋊Q8), (C6×C12).188C22, (C2×C62).45C22, C6.110(D4⋊2S3), (C22×Dic3).4C6, (C6×Dic3).91C22, C2.6(C3×S3×D4), (C2×C6)⋊2(C3×Q8), (C2×C4).5(S3×C6), C3⋊1(C3×C22⋊Q8), (C2×C12).1(C2×C6), C6.20(C3×C4○D4), C22.39(S3×C2×C6), (C3×C6).47(C2×Q8), (C3×C4⋊Dic3)⋊26C2, C2.6(C3×D4⋊2S3), (C3×C6).203(C2×D4), (C3×C22⋊C4).2C6, C22⋊C4.1(C3×S3), (Dic3×C2×C6).11C2, (C3×Dic3⋊C4)⋊23C2, (C3×C22⋊C4).15S3, (C22×C6).19(C2×C6), (C2×C6).24(C22×C6), (C2×Dic3).5(C2×C6), (C3×C6).126(C4○D4), (C2×C6).302(C22×S3), (C3×C6.D4).7C2, (C32×C22⋊C4).4C2, SmallGroup(288,649)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×Dic3.D4
G = < a,b,c,d,e | a3=b6=d4=e2=1, c2=b3, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, be=eb, dcd-1=b3c, ce=ec, ede=b3d-1 >
Subgroups: 354 in 165 conjugacy classes, 70 normal (58 characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, C22, C6, C6, C2×C4, C2×C4, Q8, C23, C32, Dic3, Dic3, C12, C2×C6, C2×C6, C2×C6, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C2×Q8, C3×C6, C3×C6, Dic6, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C3×Q8, C22×C6, C22×C6, C22⋊Q8, C3×Dic3, C3×Dic3, C3×C12, C62, C62, C62, Dic3⋊C4, C4⋊Dic3, C6.D4, C3×C22⋊C4, C3×C22⋊C4, C3×C4⋊C4, C2×Dic6, C22×Dic3, C22×C12, C6×Q8, C3×Dic6, C6×Dic3, C6×Dic3, C6×C12, C2×C62, Dic3.D4, C3×C22⋊Q8, C3×Dic3⋊C4, C3×C4⋊Dic3, C3×C6.D4, C32×C22⋊C4, C6×Dic6, Dic3×C2×C6, C3×Dic3.D4
Quotients: C1, C2, C3, C22, S3, C6, D4, Q8, C23, D6, C2×C6, C2×D4, C2×Q8, C4○D4, C3×S3, Dic6, C3×D4, C3×Q8, C22×S3, C22×C6, C22⋊Q8, S3×C6, C2×Dic6, S3×D4, D4⋊2S3, C6×D4, C6×Q8, C3×C4○D4, C3×Dic6, S3×C2×C6, Dic3.D4, C3×C22⋊Q8, C6×Dic6, C3×S3×D4, C3×D4⋊2S3, C3×Dic3.D4
►On 48 points
(1 5 3)(2 6 4)(7 9 11)(8 10 12)(13 17 15)(14 18 16)(19 23 21)(20 24 22)(25 29 27)(26 30 28)(31 33 35)(32 34 36)(37 39 41)(38 40 42)(43 45 47)(44 46 48)
(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 4 40)(2 42 5 39)(3 41 6 38)(7 22 10 19)(8 21 11 24)(9 20 12 23)(13 33 16 36)(14 32 17 35)(15 31 18 34)(25 46 28 43)(26 45 29 48)(27 44 30 47)
(1 22 14 27)(2 23 15 28)(3 24 16 29)(4 19 17 30)(5 20 18 25)(6 21 13 26)(7 32 47 37)(8 33 48 38)(9 34 43 39)(10 35 44 40)(11 36 45 41)(12 31 46 42)
(1 4)(2 5)(3 6)(7 47)(8 48)(9 43)(10 44)(11 45)(12 46)(13 16)(14 17)(15 18)(19 30)(20 25)(21 26)(22 27)(23 28)(24 29)(31 34)(32 35)(33 36)(37 40)(38 41)(39 42)
G:=sub<Sym(48)| (1,5,3)(2,6,4)(7,9,11)(8,10,12)(13,17,15)(14,18,16)(19,23,21)(20,24,22)(25,29,27)(26,30,28)(31,33,35)(32,34,36)(37,39,41)(38,40,42)(43,45,47)(44,46,48), (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), (1,37,4,40)(2,42,5,39)(3,41,6,38)(7,22,10,19)(8,21,11,24)(9,20,12,23)(13,33,16,36)(14,32,17,35)(15,31,18,34)(25,46,28,43)(26,45,29,48)(27,44,30,47), (1,22,14,27)(2,23,15,28)(3,24,16,29)(4,19,17,30)(5,20,18,25)(6,21,13,26)(7,32,47,37)(8,33,48,38)(9,34,43,39)(10,35,44,40)(11,36,45,41)(12,31,46,42), (1,4)(2,5)(3,6)(7,47)(8,48)(9,43)(10,44)(11,45)(12,46)(13,16)(14,17)(15,18)(19,30)(20,25)(21,26)(22,27)(23,28)(24,29)(31,34)(32,35)(33,36)(37,40)(38,41)(39,42)>;
G:=Group( (1,5,3)(2,6,4)(7,9,11)(8,10,12)(13,17,15)(14,18,16)(19,23,21)(20,24,22)(25,29,27)(26,30,28)(31,33,35)(32,34,36)(37,39,41)(38,40,42)(43,45,47)(44,46,48), (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), (1,37,4,40)(2,42,5,39)(3,41,6,38)(7,22,10,19)(8,21,11,24)(9,20,12,23)(13,33,16,36)(14,32,17,35)(15,31,18,34)(25,46,28,43)(26,45,29,48)(27,44,30,47), (1,22,14,27)(2,23,15,28)(3,24,16,29)(4,19,17,30)(5,20,18,25)(6,21,13,26)(7,32,47,37)(8,33,48,38)(9,34,43,39)(10,35,44,40)(11,36,45,41)(12,31,46,42), (1,4)(2,5)(3,6)(7,47)(8,48)(9,43)(10,44)(11,45)(12,46)(13,16)(14,17)(15,18)(19,30)(20,25)(21,26)(22,27)(23,28)(24,29)(31,34)(32,35)(33,36)(37,40)(38,41)(39,42) );
G=PermutationGroup([[(1,5,3),(2,6,4),(7,9,11),(8,10,12),(13,17,15),(14,18,16),(19,23,21),(20,24,22),(25,29,27),(26,30,28),(31,33,35),(32,34,36),(37,39,41),(38,40,42),(43,45,47),(44,46,48)], [(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)], [(1,37,4,40),(2,42,5,39),(3,41,6,38),(7,22,10,19),(8,21,11,24),(9,20,12,23),(13,33,16,36),(14,32,17,35),(15,31,18,34),(25,46,28,43),(26,45,29,48),(27,44,30,47)], [(1,22,14,27),(2,23,15,28),(3,24,16,29),(4,19,17,30),(5,20,18,25),(6,21,13,26),(7,32,47,37),(8,33,48,38),(9,34,43,39),(10,35,44,40),(11,36,45,41),(12,31,46,42)], [(1,4),(2,5),(3,6),(7,47),(8,48),(9,43),(10,44),(11,45),(12,46),(13,16),(14,17),(15,18),(19,30),(20,25),(21,26),(22,27),(23,28),(24,29),(31,34),(32,35),(33,36),(37,40),(38,41),(39,42)]])
72 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | ··· | 6F | 6G | ··· | 6S | 6T | ··· | 6Y | 12A | ··· | 12P | 12Q | ··· | 12X | 12Y | 12Z | 12AA | 12AB |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | ··· | 12 | 12 | ··· | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 6 | 6 | 6 | 6 | 12 | 12 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 12 | 12 | 12 | 12 |
72 irreducible representations
| dim | 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 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | + | + | - | + | - | |||||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | C6 | C6 | S3 | D4 | Q8 | D6 | D6 | C4○D4 | C3×S3 | C3×D4 | Dic6 | C3×Q8 | S3×C6 | S3×C6 | C3×C4○D4 | C3×Dic6 | S3×D4 | D4⋊2S3 | C3×S3×D4 | C3×D4⋊2S3 |
| kernel | C3×Dic3.D4 | C3×Dic3⋊C4 | C3×C4⋊Dic3 | C3×C6.D4 | C32×C22⋊C4 | C6×Dic6 | Dic3×C2×C6 | Dic3.D4 | Dic3⋊C4 | C4⋊Dic3 | C6.D4 | C3×C22⋊C4 | C2×Dic6 | C22×Dic3 | C3×C22⋊C4 | C3×Dic3 | C62 | C2×C12 | C22×C6 | C3×C6 | C22⋊C4 | Dic3 | C2×C6 | C2×C6 | C2×C4 | C23 | C6 | C22 | C6 | C6 | C2 | C2 |
| # reps | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 2 | 2 | 2 | 2 | 2 | 1 | 2 | 2 | 2 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | 4 | 8 | 1 | 1 | 2 | 2 |
Matrix representation of C3×Dic3.D4 ►in GL4(𝔽13) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 3 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 10 | 0 |
| 0 | 0 | 9 | 4 |
| 12 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 1 | 8 |
| 0 | 0 | 3 | 12 |
| 0 | 12 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 5 | 0 |
| 0 | 0 | 2 | 8 |
| 1 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
G:=sub<GL(4,GF(13))| [1,0,0,0,0,1,0,0,0,0,3,0,0,0,0,3],[1,0,0,0,0,1,0,0,0,0,10,9,0,0,0,4],[12,0,0,0,0,12,0,0,0,0,1,3,0,0,8,12],[0,1,0,0,12,0,0,0,0,0,5,2,0,0,0,8],[1,0,0,0,0,12,0,0,0,0,12,0,0,0,0,12] >;
C3×Dic3.D4 in GAP, Magma, Sage, TeX
C_3\times {\rm Dic}_3.D_4 % in TeX
G:=Group("C3xDic3.D4"); // GroupNames label
G:=SmallGroup(288,649);
// by ID
G=gap.SmallGroup(288,649);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,168,590,555,142,9414]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^6=d^4=e^2=1,c^2=b^3,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=b^-1,b*d=d*b,b*e=e*b,d*c*d^-1=b^3*c,c*e=e*c,e*d*e=b^3*d^-1>;
// generators/relations