direct product, metabelian, supersoluble, monomial
Aliases: C3×D4.Dic3, D4.(C3×Dic3), (C3×D4).1C12, C4.Dic3⋊8C6, (C3×Q8).5C12, C4.5(C6×Dic3), C12.15(C2×C12), (C2×C12).332D6, C32⋊13(C8○D4), C62.61(C2×C4), (C3×D4).6Dic3, Q8.3(C3×Dic3), (D4×C32).2C4, (Q8×C32).2C4, C6.27(C22×C12), C12.42(C22×C6), (C3×Q8).10Dic3, C12.40(C2×Dic3), C22.1(C6×Dic3), C12.230(C22×S3), (C6×C12).130C22, (C3×C12).174C23, C6.47(C22×Dic3), (C2×C3⋊C8)⋊7C6, (C6×C3⋊C8)⋊26C2, C3⋊3(C3×C8○D4), C4.42(S3×C2×C6), C3⋊C8.13(C2×C6), C2.8(Dic3×C2×C6), (C2×C6).7(C2×C12), (C2×C4).58(S3×C6), C4○D4.7(C3×S3), (C3×C12).70(C2×C4), (C2×C12).41(C2×C6), (C3×C4○D4).10C6, (C3×C4○D4).22S3, (C3×C3⋊C8).44C22, (C3×C4.Dic3)⋊22C2, (C2×C6).11(C2×Dic3), (C32×C4○D4).3C2, (C3×C6).118(C22×C4), SmallGroup(288,719)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×D4.Dic3
G = < a,b,c,d,e | a3=b4=c2=1, d6=b2, e2=b2d3, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=d5 >
Subgroups: 226 in 144 conjugacy classes, 90 normal (24 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C6, C6, C8, C2×C4, D4, Q8, C32, C12, C12, C12, C2×C6, C2×C6, C2×C8, M4(2), C4○D4, C3×C6, C3×C6, C3⋊C8, C3⋊C8, C24, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C3×Q8, C8○D4, C3×C12, C3×C12, C62, C2×C3⋊C8, C4.Dic3, C2×C24, C3×M4(2), C3×C4○D4, C3×C4○D4, C3×C3⋊C8, C3×C3⋊C8, C6×C12, D4×C32, Q8×C32, D4.Dic3, C3×C8○D4, C6×C3⋊C8, C3×C4.Dic3, C32×C4○D4, C3×D4.Dic3
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, C23, Dic3, C12, D6, C2×C6, C22×C4, C3×S3, C2×Dic3, C2×C12, C22×S3, C22×C6, C8○D4, C3×Dic3, S3×C6, C22×Dic3, C22×C12, C6×Dic3, S3×C2×C6, D4.Dic3, C3×C8○D4, Dic3×C2×C6, C3×D4.Dic3
►On 48 points
(1 5 9)(2 6 10)(3 7 11)(4 8 12)(13 17 21)(14 18 22)(15 19 23)(16 20 24)(25 33 29)(26 34 30)(27 35 31)(28 36 32)(37 45 41)(38 46 42)(39 47 43)(40 48 44)
(1 10 7 4)(2 11 8 5)(3 12 9 6)(13 16 19 22)(14 17 20 23)(15 18 21 24)(25 34 31 28)(26 35 32 29)(27 36 33 30)(37 40 43 46)(38 41 44 47)(39 42 45 48)
(1 13)(2 14)(3 15)(4 16)(5 17)(6 18)(7 19)(8 20)(9 21)(10 22)(11 23)(12 24)(25 46)(26 47)(27 48)(28 37)(29 38)(30 39)(31 40)(32 41)(33 42)(34 43)(35 44)(36 45)
(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 10 27 7 36 4 33)(2 35 11 32 8 29 5 26)(3 28 12 25 9 34 6 31)(13 39 22 48 19 45 16 42)(14 44 23 41 20 38 17 47)(15 37 24 46 21 43 18 40)
G:=sub<Sym(48)| (1,5,9)(2,6,10)(3,7,11)(4,8,12)(13,17,21)(14,18,22)(15,19,23)(16,20,24)(25,33,29)(26,34,30)(27,35,31)(28,36,32)(37,45,41)(38,46,42)(39,47,43)(40,48,44), (1,10,7,4)(2,11,8,5)(3,12,9,6)(13,16,19,22)(14,17,20,23)(15,18,21,24)(25,34,31,28)(26,35,32,29)(27,36,33,30)(37,40,43,46)(38,41,44,47)(39,42,45,48), (1,13)(2,14)(3,15)(4,16)(5,17)(6,18)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24)(25,46)(26,47)(27,48)(28,37)(29,38)(30,39)(31,40)(32,41)(33,42)(34,43)(35,44)(36,45), (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), (1,30,10,27,7,36,4,33)(2,35,11,32,8,29,5,26)(3,28,12,25,9,34,6,31)(13,39,22,48,19,45,16,42)(14,44,23,41,20,38,17,47)(15,37,24,46,21,43,18,40)>;
G:=Group( (1,5,9)(2,6,10)(3,7,11)(4,8,12)(13,17,21)(14,18,22)(15,19,23)(16,20,24)(25,33,29)(26,34,30)(27,35,31)(28,36,32)(37,45,41)(38,46,42)(39,47,43)(40,48,44), (1,10,7,4)(2,11,8,5)(3,12,9,6)(13,16,19,22)(14,17,20,23)(15,18,21,24)(25,34,31,28)(26,35,32,29)(27,36,33,30)(37,40,43,46)(38,41,44,47)(39,42,45,48), (1,13)(2,14)(3,15)(4,16)(5,17)(6,18)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24)(25,46)(26,47)(27,48)(28,37)(29,38)(30,39)(31,40)(32,41)(33,42)(34,43)(35,44)(36,45), (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), (1,30,10,27,7,36,4,33)(2,35,11,32,8,29,5,26)(3,28,12,25,9,34,6,31)(13,39,22,48,19,45,16,42)(14,44,23,41,20,38,17,47)(15,37,24,46,21,43,18,40) );
G=PermutationGroup([[(1,5,9),(2,6,10),(3,7,11),(4,8,12),(13,17,21),(14,18,22),(15,19,23),(16,20,24),(25,33,29),(26,34,30),(27,35,31),(28,36,32),(37,45,41),(38,46,42),(39,47,43),(40,48,44)], [(1,10,7,4),(2,11,8,5),(3,12,9,6),(13,16,19,22),(14,17,20,23),(15,18,21,24),(25,34,31,28),(26,35,32,29),(27,36,33,30),(37,40,43,46),(38,41,44,47),(39,42,45,48)], [(1,13),(2,14),(3,15),(4,16),(5,17),(6,18),(7,19),(8,20),(9,21),(10,22),(11,23),(12,24),(25,46),(26,47),(27,48),(28,37),(29,38),(30,39),(31,40),(32,41),(33,42),(34,43),(35,44),(36,45)], [(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)], [(1,30,10,27,7,36,4,33),(2,35,11,32,8,29,5,26),(3,28,12,25,9,34,6,31),(13,39,22,48,19,45,16,42),(14,44,23,41,20,38,17,47),(15,37,24,46,21,43,18,40)]])
90 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 6C | ··· | 6K | 6L | ··· | 6T | 8A | 8B | 8C | 8D | 8E | ··· | 8J | 12A | 12B | 12C | 12D | 12E | ··· | 12P | 12Q | ··· | 12Y | 24A | ··· | 24H | 24I | ··· | 24T |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | ··· | 6 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 8 | ··· | 8 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 12 | ··· | 12 | 24 | ··· | 24 | 24 | ··· | 24 |
| size | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 3 | 3 | 3 | 3 | 6 | ··· | 6 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 3 | ··· | 3 | 6 | ··· | 6 |
90 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | - | - | ||||||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C4 | C4 | C6 | C6 | C6 | C12 | C12 | S3 | D6 | Dic3 | Dic3 | C3×S3 | C8○D4 | S3×C6 | C3×Dic3 | C3×Dic3 | C3×C8○D4 | D4.Dic3 | C3×D4.Dic3 |
| kernel | C3×D4.Dic3 | C6×C3⋊C8 | C3×C4.Dic3 | C32×C4○D4 | D4.Dic3 | D4×C32 | Q8×C32 | C2×C3⋊C8 | C4.Dic3 | C3×C4○D4 | C3×D4 | C3×Q8 | C3×C4○D4 | C2×C12 | C3×D4 | C3×Q8 | C4○D4 | C32 | C2×C4 | D4 | Q8 | C3 | C3 | C1 |
| # reps | 1 | 3 | 3 | 1 | 2 | 6 | 2 | 6 | 6 | 2 | 12 | 4 | 1 | 3 | 3 | 1 | 2 | 4 | 6 | 6 | 2 | 8 | 2 | 4 |
Matrix representation of C3×D4.Dic3 ►in GL4(𝔽73) generated by
| 64 | 0 | 0 | 0 |
| 0 | 64 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 72 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 46 | 0 |
| 0 | 0 | 43 | 27 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 27 | 39 |
| 0 | 0 | 30 | 46 |
| 9 | 0 | 0 | 0 |
| 0 | 65 | 0 | 0 |
| 0 | 0 | 46 | 0 |
| 0 | 0 | 0 | 46 |
| 0 | 1 | 0 | 0 |
| 72 | 0 | 0 | 0 |
| 0 | 0 | 22 | 0 |
| 0 | 0 | 0 | 22 |
G:=sub<GL(4,GF(73))| [64,0,0,0,0,64,0,0,0,0,1,0,0,0,0,1],[72,0,0,0,0,72,0,0,0,0,46,43,0,0,0,27],[1,0,0,0,0,1,0,0,0,0,27,30,0,0,39,46],[9,0,0,0,0,65,0,0,0,0,46,0,0,0,0,46],[0,72,0,0,1,0,0,0,0,0,22,0,0,0,0,22] >;
C3×D4.Dic3 in GAP, Magma, Sage, TeX
C_3\times D_4.{\rm Dic}_3 % in TeX
G:=Group("C3xD4.Dic3"); // GroupNames label
G:=SmallGroup(288,719);
// by ID
G=gap.SmallGroup(288,719);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,168,555,102,9414]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^4=c^2=1,d^6=b^2,e^2=b^2*d^3,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^5>;
// generators/relations