direct product, metabelian, supersoluble, monomial
Aliases: C3×Dic3⋊4D4, C62.172C23, C3⋊D4⋊C12, D6⋊C4⋊9C6, C3⋊2(D4×C12), D6⋊2(C2×C12), C6.18(C6×D4), C32⋊18(C4×D4), Dic3⋊C4⋊9C6, C22⋊3(S3×C12), C62⋊10(C2×C4), Dic3⋊4(C3×D4), C6.176(S3×D4), (C3×Dic3)⋊19D4, (C4×Dic3)⋊11C6, Dic3⋊1(C2×C12), (C2×C12).265D6, C23.24(S3×C6), C6.7(C22×C12), (Dic3×C12)⋊32C2, (C22×Dic3)⋊4C6, (C22×C6).105D6, (C6×C12).242C22, (C2×C62).48C22, C6.112(D4⋊2S3), (C6×Dic3).121C22, (S3×C2×C4)⋊9C6, C2.2(C3×S3×D4), C2.9(S3×C2×C12), (S3×C2×C12)⋊24C2, (C2×C6)⋊11(C4×S3), (C2×C6)⋊4(C2×C12), (C3×C3⋊D4)⋊3C4, C6.106(S3×C2×C4), (Dic3×C2×C6)⋊5C2, (S3×C6)⋊15(C2×C4), (C3×D6⋊C4)⋊27C2, C22⋊C4⋊7(C3×S3), (C3×C22⋊C4)⋊9C6, (C2×C4).26(S3×C6), C6.21(C3×C4○D4), (C6×C3⋊D4).9C2, (C2×C3⋊D4).2C6, C22.14(S3×C2×C6), (C3×C22⋊C4)⋊15S3, (C2×C12).51(C2×C6), C2.2(C3×D4⋊2S3), (C3×C6).205(C2×D4), (S3×C2×C6).89C22, (C3×Dic3⋊C4)⋊28C2, (C3×Dic3)⋊10(C2×C4), (C22×C6).22(C2×C6), (C2×C6).27(C22×C6), (C3×C6).78(C22×C4), (C3×C6).128(C4○D4), (C32×C22⋊C4)⋊15C2, (C22×S3).17(C2×C6), (C2×C6).305(C22×S3), (C2×Dic3).20(C2×C6), SmallGroup(288,652)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×Dic3⋊4D4
G = < a,b,c,d,e | a3=b6=d4=e2=1, c2=b3, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=dbd-1=b-1, be=eb, cd=dc, ce=ec, ede=d-1 >
Subgroups: 450 in 205 conjugacy classes, 86 normal (58 characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, C23, C23, C32, Dic3, Dic3, C12, D6, D6, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C3×S3, C3×C6, C3×C6, C4×S3, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C22×S3, C22×C6, C22×C6, C4×D4, C3×Dic3, C3×Dic3, C3×C12, S3×C6, S3×C6, C62, C62, C62, C4×Dic3, Dic3⋊C4, D6⋊C4, C4×C12, C3×C22⋊C4, C3×C22⋊C4, C3×C4⋊C4, S3×C2×C4, C22×Dic3, C2×C3⋊D4, C22×C12, C6×D4, S3×C12, C6×Dic3, C6×Dic3, C3×C3⋊D4, C6×C12, S3×C2×C6, C2×C62, Dic3⋊4D4, D4×C12, Dic3×C12, C3×Dic3⋊C4, C3×D6⋊C4, C32×C22⋊C4, S3×C2×C12, Dic3×C2×C6, C6×C3⋊D4, C3×Dic3⋊4D4
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, D4, C23, C12, D6, C2×C6, C22×C4, C2×D4, C4○D4, C3×S3, C4×S3, C2×C12, C3×D4, C22×S3, C22×C6, C4×D4, S3×C6, S3×C2×C4, S3×D4, D4⋊2S3, C22×C12, C6×D4, C3×C4○D4, S3×C12, S3×C2×C6, Dic3⋊4D4, D4×C12, S3×C2×C12, C3×S3×D4, C3×D4⋊2S3, C3×Dic3⋊4D4
►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 45 4 48)(2 44 5 47)(3 43 6 46)(7 17 10 14)(8 16 11 13)(9 15 12 18)(19 35 22 32)(20 34 23 31)(21 33 24 36)(25 39 28 42)(26 38 29 41)(27 37 30 40)
(1 36 17 40)(2 35 18 39)(3 34 13 38)(4 33 14 37)(5 32 15 42)(6 31 16 41)(7 30 48 24)(8 29 43 23)(9 28 44 22)(10 27 45 21)(11 26 46 20)(12 25 47 19)
(19 25)(20 26)(21 27)(22 28)(23 29)(24 30)(31 41)(32 42)(33 37)(34 38)(35 39)(36 40)
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), (1,45,4,48)(2,44,5,47)(3,43,6,46)(7,17,10,14)(8,16,11,13)(9,15,12,18)(19,35,22,32)(20,34,23,31)(21,33,24,36)(25,39,28,42)(26,38,29,41)(27,37,30,40), (1,36,17,40)(2,35,18,39)(3,34,13,38)(4,33,14,37)(5,32,15,42)(6,31,16,41)(7,30,48,24)(8,29,43,23)(9,28,44,22)(10,27,45,21)(11,26,46,20)(12,25,47,19), (19,25)(20,26)(21,27)(22,28)(23,29)(24,30)(31,41)(32,42)(33,37)(34,38)(35,39)(36,40)>;
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), (1,45,4,48)(2,44,5,47)(3,43,6,46)(7,17,10,14)(8,16,11,13)(9,15,12,18)(19,35,22,32)(20,34,23,31)(21,33,24,36)(25,39,28,42)(26,38,29,41)(27,37,30,40), (1,36,17,40)(2,35,18,39)(3,34,13,38)(4,33,14,37)(5,32,15,42)(6,31,16,41)(7,30,48,24)(8,29,43,23)(9,28,44,22)(10,27,45,21)(11,26,46,20)(12,25,47,19), (19,25)(20,26)(21,27)(22,28)(23,29)(24,30)(31,41)(32,42)(33,37)(34,38)(35,39)(36,40) );
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)], [(1,45,4,48),(2,44,5,47),(3,43,6,46),(7,17,10,14),(8,16,11,13),(9,15,12,18),(19,35,22,32),(20,34,23,31),(21,33,24,36),(25,39,28,42),(26,38,29,41),(27,37,30,40)], [(1,36,17,40),(2,35,18,39),(3,34,13,38),(4,33,14,37),(5,32,15,42),(6,31,16,41),(7,30,48,24),(8,29,43,23),(9,28,44,22),(10,27,45,21),(11,26,46,20),(12,25,47,19)], [(19,25),(20,26),(21,27),(22,28),(23,29),(24,30),(31,41),(32,42),(33,37),(34,38),(35,39),(36,40)]])
90 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 6A | ··· | 6F | 6G | ··· | 6S | 6T | ··· | 6Y | 6Z | 6AA | 6AB | 6AC | 12A | ··· | 12H | 12I | ··· | 12P | 12Q | ··· | 12AB | 12AC | ··· | 12AJ |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 12 | ··· | 12 | 12 | ··· | 12 | 12 | ··· | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 6 | 6 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 6 | 6 | 6 | 6 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | 2 | ··· | 2 | 3 | ··· | 3 | 4 | ··· | 4 | 6 | ··· | 6 |
90 irreducible representations
| dim | 1 | 1 | 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 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | ||||||||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C3 | C4 | C6 | C6 | C6 | C6 | C6 | C6 | C6 | C12 | S3 | D4 | D6 | D6 | C4○D4 | C3×S3 | C3×D4 | C4×S3 | S3×C6 | S3×C6 | C3×C4○D4 | S3×C12 | S3×D4 | D4⋊2S3 | C3×S3×D4 | C3×D4⋊2S3 |
| kernel | C3×Dic3⋊4D4 | Dic3×C12 | C3×Dic3⋊C4 | C3×D6⋊C4 | C32×C22⋊C4 | S3×C2×C12 | Dic3×C2×C6 | C6×C3⋊D4 | Dic3⋊4D4 | C3×C3⋊D4 | C4×Dic3 | Dic3⋊C4 | D6⋊C4 | C3×C22⋊C4 | S3×C2×C4 | C22×Dic3 | C2×C3⋊D4 | C3⋊D4 | C3×C22⋊C4 | C3×Dic3 | C2×C12 | C22×C6 | C3×C6 | C22⋊C4 | Dic3 | C2×C6 | C2×C4 | C23 | C6 | C22 | C6 | C6 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 8 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 16 | 1 | 2 | 2 | 1 | 2 | 2 | 4 | 4 | 4 | 2 | 4 | 8 | 1 | 1 | 2 | 2 |
Matrix representation of C3×Dic3⋊4D4 ►in GL4(𝔽13) generated by
| 9 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 4 | 0 | 0 | 0 |
| 10 | 10 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 5 | 3 | 0 | 0 |
| 0 | 8 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 12 | 2 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 5 | 2 |
| 0 | 0 | 0 | 8 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 8 | 12 |
G:=sub<GL(4,GF(13))| [9,0,0,0,0,9,0,0,0,0,1,0,0,0,0,1],[4,10,0,0,0,10,0,0,0,0,1,0,0,0,0,1],[5,0,0,0,3,8,0,0,0,0,1,0,0,0,0,1],[12,0,0,0,2,1,0,0,0,0,5,0,0,0,2,8],[1,0,0,0,0,1,0,0,0,0,1,8,0,0,0,12] >;
C3×Dic3⋊4D4 in GAP, Magma, Sage, TeX
C_3\times {\rm Dic}_3\rtimes_4D_4 % in TeX
G:=Group("C3xDic3:4D4"); // GroupNames label
G:=SmallGroup(288,652);
// by ID
G=gap.SmallGroup(288,652);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,701,555,142,9414]);
// 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=d*b*d^-1=b^-1,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations