direct product, metabelian, supersoluble, monomial
Aliases: C3×D4⋊Dic3, C62.107D4, (C3×D4)⋊1C12, (C6×D4).1C6, C12.7(C3×D4), (C3×C6).34D8, C6.12(C3×D8), C12.7(C2×C12), C4⋊Dic3⋊10C6, D4⋊1(C3×Dic3), (C3×D4)⋊4Dic3, (C6×D4).22S3, (C3×C12).45D4, (D4×C32)⋊4C4, C4.1(C6×Dic3), C6.5(C3×SD16), C6.34(D4⋊S3), (C2×C12).317D6, (C3×C6).22SD16, C12.99(C3⋊D4), (C6×C12).47C22, C6.16(D4.S3), C12.36(C2×Dic3), C32⋊12(D4⋊C4), C6.31(C6.D4), (C6×C3⋊C8)⋊8C2, (C2×C3⋊C8)⋊2C6, (D4×C3×C6).1C2, C2.3(C3×D4⋊S3), C3⋊3(C3×D4⋊C4), (C2×C4).38(S3×C6), (C3×C4⋊Dic3)⋊3C2, (C2×D4).1(C3×S3), (C2×C6).42(C3×D4), C4.12(C3×C3⋊D4), C2.3(C3×D4.S3), (C3×C12).43(C2×C4), (C2×C12).17(C2×C6), C6.13(C3×C22⋊C4), C2.3(C3×C6.D4), C22.17(C3×C3⋊D4), (C2×C6).110(C3⋊D4), (C3×C6).64(C22⋊C4), SmallGroup(288,266)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×D4⋊Dic3
G = < a,b,c,d,e | a3=b4=c2=d6=1, e2=d3, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe-1=b-1, bd=db, cd=dc, ece-1=bc, ede-1=d-1 >
Subgroups: 298 in 127 conjugacy classes, 50 normal (42 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, C6, C6, C8, C2×C4, C2×C4, D4, D4, C23, C32, Dic3, C12, C12, C2×C6, C2×C6, C4⋊C4, C2×C8, C2×D4, C3×C6, C3×C6, C3⋊C8, C24, C2×Dic3, C2×C12, C2×C12, C3×D4, C3×D4, C22×C6, D4⋊C4, C3×Dic3, C3×C12, C62, C62, C2×C3⋊C8, C4⋊Dic3, C3×C4⋊C4, C2×C24, C6×D4, C6×D4, C3×C3⋊C8, C6×Dic3, C6×C12, D4×C32, D4×C32, C2×C62, D4⋊Dic3, C3×D4⋊C4, C6×C3⋊C8, C3×C4⋊Dic3, D4×C3×C6, C3×D4⋊Dic3
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, D4, Dic3, C12, D6, C2×C6, C22⋊C4, D8, SD16, C3×S3, C2×Dic3, C3⋊D4, C2×C12, C3×D4, D4⋊C4, C3×Dic3, S3×C6, D4⋊S3, D4.S3, C6.D4, C3×C22⋊C4, C3×D8, C3×SD16, C6×Dic3, C3×C3⋊D4, D4⋊Dic3, C3×D4⋊C4, C3×D4⋊S3, C3×D4.S3, C3×C6.D4, C3×D4⋊Dic3
►On 48 points
(1 3 5)(2 4 6)(7 9 11)(8 10 12)(13 17 15)(14 18 16)(19 21 23)(20 22 24)(25 27 29)(26 28 30)(31 35 33)(32 36 34)(37 41 39)(38 42 40)(43 47 45)(44 48 46)
(1 30 19 7)(2 25 20 8)(3 26 21 9)(4 27 22 10)(5 28 23 11)(6 29 24 12)(13 32 42 47)(14 33 37 48)(15 34 38 43)(16 35 39 44)(17 36 40 45)(18 31 41 46)
(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)(19 30)(20 25)(21 26)(22 27)(23 28)(24 29)(31 46)(32 47)(33 48)(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 35 4 32)(2 34 5 31)(3 33 6 36)(7 39 10 42)(8 38 11 41)(9 37 12 40)(13 30 16 27)(14 29 17 26)(15 28 18 25)(19 44 22 47)(20 43 23 46)(21 48 24 45)
G:=sub<Sym(48)| (1,3,5)(2,4,6)(7,9,11)(8,10,12)(13,17,15)(14,18,16)(19,21,23)(20,22,24)(25,27,29)(26,28,30)(31,35,33)(32,36,34)(37,41,39)(38,42,40)(43,47,45)(44,48,46), (1,30,19,7)(2,25,20,8)(3,26,21,9)(4,27,22,10)(5,28,23,11)(6,29,24,12)(13,32,42,47)(14,33,37,48)(15,34,38,43)(16,35,39,44)(17,36,40,45)(18,31,41,46), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(19,30)(20,25)(21,26)(22,27)(23,28)(24,29)(31,46)(32,47)(33,48)(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), (1,35,4,32)(2,34,5,31)(3,33,6,36)(7,39,10,42)(8,38,11,41)(9,37,12,40)(13,30,16,27)(14,29,17,26)(15,28,18,25)(19,44,22,47)(20,43,23,46)(21,48,24,45)>;
G:=Group( (1,3,5)(2,4,6)(7,9,11)(8,10,12)(13,17,15)(14,18,16)(19,21,23)(20,22,24)(25,27,29)(26,28,30)(31,35,33)(32,36,34)(37,41,39)(38,42,40)(43,47,45)(44,48,46), (1,30,19,7)(2,25,20,8)(3,26,21,9)(4,27,22,10)(5,28,23,11)(6,29,24,12)(13,32,42,47)(14,33,37,48)(15,34,38,43)(16,35,39,44)(17,36,40,45)(18,31,41,46), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(19,30)(20,25)(21,26)(22,27)(23,28)(24,29)(31,46)(32,47)(33,48)(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), (1,35,4,32)(2,34,5,31)(3,33,6,36)(7,39,10,42)(8,38,11,41)(9,37,12,40)(13,30,16,27)(14,29,17,26)(15,28,18,25)(19,44,22,47)(20,43,23,46)(21,48,24,45) );
G=PermutationGroup([[(1,3,5),(2,4,6),(7,9,11),(8,10,12),(13,17,15),(14,18,16),(19,21,23),(20,22,24),(25,27,29),(26,28,30),(31,35,33),(32,36,34),(37,41,39),(38,42,40),(43,47,45),(44,48,46)], [(1,30,19,7),(2,25,20,8),(3,26,21,9),(4,27,22,10),(5,28,23,11),(6,29,24,12),(13,32,42,47),(14,33,37,48),(15,34,38,43),(16,35,39,44),(17,36,40,45),(18,31,41,46)], [(1,7),(2,8),(3,9),(4,10),(5,11),(6,12),(19,30),(20,25),(21,26),(22,27),(23,28),(24,29),(31,46),(32,47),(33,48),(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)], [(1,35,4,32),(2,34,5,31),(3,33,6,36),(7,39,10,42),(8,38,11,41),(9,37,12,40),(13,30,16,27),(14,29,17,26),(15,28,18,25),(19,44,22,47),(20,43,23,46),(21,48,24,45)]])
72 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 4C | 4D | 6A | ··· | 6F | 6G | ··· | 6O | 6P | ··· | 6AE | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | ··· | 12J | 12K | 12L | 12M | 12N | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 12 | 12 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 12 | 12 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 12 | 12 | 12 | 12 | 6 | ··· | 6 |
72 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 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | + | - | ||||||||||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C4 | C6 | C6 | C6 | C12 | S3 | D4 | D4 | D6 | Dic3 | D8 | SD16 | C3×S3 | C3⋊D4 | C3×D4 | C3⋊D4 | C3×D4 | S3×C6 | C3×Dic3 | C3×D8 | C3×SD16 | C3×C3⋊D4 | C3×C3⋊D4 | D4⋊S3 | D4.S3 | C3×D4⋊S3 | C3×D4.S3 |
| kernel | C3×D4⋊Dic3 | C6×C3⋊C8 | C3×C4⋊Dic3 | D4×C3×C6 | D4⋊Dic3 | D4×C32 | C2×C3⋊C8 | C4⋊Dic3 | C6×D4 | C3×D4 | C6×D4 | C3×C12 | C62 | C2×C12 | C3×D4 | C3×C6 | C3×C6 | C2×D4 | C12 | C12 | C2×C6 | C2×C6 | C2×C4 | D4 | C6 | C6 | C4 | C22 | C6 | C6 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 4 | 2 | 2 | 2 | 8 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 1 | 1 | 2 | 2 |
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 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 72 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
| 65 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 72 | 0 | 0 | 0 |
| 0 | 0 | 57 | 16 |
| 0 | 0 | 16 | 16 |
G:=sub<GL(4,GF(73))| [64,0,0,0,0,64,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,0,72,0,0,1,0],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0],[65,0,0,0,0,9,0,0,0,0,1,0,0,0,0,1],[0,72,0,0,1,0,0,0,0,0,57,16,0,0,16,16] >;
C3×D4⋊Dic3 in GAP, Magma, Sage, TeX
C_3\times D_4\rtimes {\rm Dic}_3 % in TeX
G:=Group("C3xD4:Dic3"); // GroupNames label
G:=SmallGroup(288,266);
// by ID
G=gap.SmallGroup(288,266);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-3,84,365,2524,1271,102,9414]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^4=c^2=d^6=1,e^2=d^3,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=e*b*e^-1=b^-1,b*d=d*b,c*d=d*c,e*c*e^-1=b*c,e*d*e^-1=d^-1>;
// generators/relations