direct product, metabelian, supersoluble, monomial
Aliases: C6×D4⋊2S3, C62.269C23, D4⋊5(S3×C6), (C6×D4)⋊6C6, (C6×D4)⋊17S3, (C3×D4)⋊23D6, Dic6⋊7(C2×C6), C6.6(C23×C6), (C6×Dic6)⋊22C2, (C2×Dic6)⋊12C6, (C2×C12).334D6, C23.29(S3×C6), C6.74(S3×C23), (C3×C6).43C24, D6.2(C22×C6), (S3×C12)⋊20C22, (S3×C6).29C23, C12.20(C22×C6), (C22×C6).109D6, C12.171(C22×S3), (C6×C12).163C22, (C3×C12).121C23, (C3×Dic6)⋊32C22, (C6×Dic3)⋊34C22, (C22×Dic3)⋊11C6, (D4×C32)⋊19C22, (C2×C62).83C22, Dic3.3(C22×C6), (C3×Dic3).30C23, (S3×C2×C4)⋊4C6, (D4×C3×C6)⋊10C2, C3⋊2(C6×C4○D4), C6⋊2(C3×C4○D4), C4.20(S3×C2×C6), (S3×C2×C12)⋊12C2, (C4×S3)⋊4(C2×C6), (C2×D4)⋊8(C3×S3), (C3×D4)⋊6(C2×C6), C3⋊D4⋊2(C2×C6), C2.7(S3×C22×C6), C22.1(S3×C2×C6), (C3×C6)⋊9(C4○D4), (C6×C3⋊D4)⋊24C2, (C2×C3⋊D4)⋊10C6, (C2×C4).60(S3×C6), (Dic3×C2×C6)⋊19C2, C32⋊15(C2×C4○D4), (C2×C12).45(C2×C6), (C2×Dic3)⋊9(C2×C6), (C2×C6).1(C22×C6), (C3×C3⋊D4)⋊16C22, (S3×C2×C6).110C22, (C22×C6).34(C2×C6), (C2×C6).20(C22×S3), (C22×S3).30(C2×C6), SmallGroup(288,993)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C6×D4⋊2S3
G = < a,b,c,d,e | a6=b4=c2=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ece=b2c, ede=d-1 >
Subgroups: 682 in 355 conjugacy classes, 178 normal (30 characteristic)
C1, C2, C2, C2, C3, C3, C4, C4, C22, C22, C22, S3, C6, C6, C6, C2×C4, C2×C4, D4, D4, Q8, C23, C23, C32, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C2×C6, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C3×S3, C3×C6, C3×C6, C3×C6, Dic6, C4×S3, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C22×S3, C22×C6, C22×C6, C2×C4○D4, C3×Dic3, C3×C12, S3×C6, S3×C6, C62, C62, C62, C2×Dic6, S3×C2×C4, D4⋊2S3, C22×Dic3, C2×C3⋊D4, C22×C12, C6×D4, C6×D4, C6×Q8, C3×C4○D4, C3×Dic6, S3×C12, C6×Dic3, C6×Dic3, C3×C3⋊D4, C6×C12, D4×C32, S3×C2×C6, C2×C62, C2×D4⋊2S3, C6×C4○D4, C6×Dic6, S3×C2×C12, C3×D4⋊2S3, Dic3×C2×C6, C6×C3⋊D4, D4×C3×C6, C6×D4⋊2S3
Quotients: C1, C2, C3, C22, S3, C6, C23, D6, C2×C6, C4○D4, C24, C3×S3, C22×S3, C22×C6, C2×C4○D4, S3×C6, D4⋊2S3, C3×C4○D4, S3×C23, C23×C6, S3×C2×C6, C2×D4⋊2S3, C6×C4○D4, C3×D4⋊2S3, S3×C22×C6, C6×D4⋊2S3
►On 48 points
(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 12 23 28)(2 7 24 29)(3 8 19 30)(4 9 20 25)(5 10 21 26)(6 11 22 27)(13 33 37 46)(14 34 38 47)(15 35 39 48)(16 36 40 43)(17 31 41 44)(18 32 42 45)
(7 29)(8 30)(9 25)(10 26)(11 27)(12 28)(31 44)(32 45)(33 46)(34 47)(35 48)(36 43)
(1 5 3)(2 6 4)(7 11 9)(8 12 10)(13 15 17)(14 16 18)(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 31)(2 32)(3 33)(4 34)(5 35)(6 36)(7 42)(8 37)(9 38)(10 39)(11 40)(12 41)(13 30)(14 25)(15 26)(16 27)(17 28)(18 29)(19 46)(20 47)(21 48)(22 43)(23 44)(24 45)
G:=sub<Sym(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), (1,12,23,28)(2,7,24,29)(3,8,19,30)(4,9,20,25)(5,10,21,26)(6,11,22,27)(13,33,37,46)(14,34,38,47)(15,35,39,48)(16,36,40,43)(17,31,41,44)(18,32,42,45), (7,29)(8,30)(9,25)(10,26)(11,27)(12,28)(31,44)(32,45)(33,46)(34,47)(35,48)(36,43), (1,5,3)(2,6,4)(7,11,9)(8,12,10)(13,15,17)(14,16,18)(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), (1,31)(2,32)(3,33)(4,34)(5,35)(6,36)(7,42)(8,37)(9,38)(10,39)(11,40)(12,41)(13,30)(14,25)(15,26)(16,27)(17,28)(18,29)(19,46)(20,47)(21,48)(22,43)(23,44)(24,45)>;
G:=Group( (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), (1,12,23,28)(2,7,24,29)(3,8,19,30)(4,9,20,25)(5,10,21,26)(6,11,22,27)(13,33,37,46)(14,34,38,47)(15,35,39,48)(16,36,40,43)(17,31,41,44)(18,32,42,45), (7,29)(8,30)(9,25)(10,26)(11,27)(12,28)(31,44)(32,45)(33,46)(34,47)(35,48)(36,43), (1,5,3)(2,6,4)(7,11,9)(8,12,10)(13,15,17)(14,16,18)(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), (1,31)(2,32)(3,33)(4,34)(5,35)(6,36)(7,42)(8,37)(9,38)(10,39)(11,40)(12,41)(13,30)(14,25)(15,26)(16,27)(17,28)(18,29)(19,46)(20,47)(21,48)(22,43)(23,44)(24,45) );
G=PermutationGroup([[(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)], [(1,12,23,28),(2,7,24,29),(3,8,19,30),(4,9,20,25),(5,10,21,26),(6,11,22,27),(13,33,37,46),(14,34,38,47),(15,35,39,48),(16,36,40,43),(17,31,41,44),(18,32,42,45)], [(7,29),(8,30),(9,25),(10,26),(11,27),(12,28),(31,44),(32,45),(33,46),(34,47),(35,48),(36,43)], [(1,5,3),(2,6,4),(7,11,9),(8,12,10),(13,15,17),(14,16,18),(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)], [(1,31),(2,32),(3,33),(4,34),(5,35),(6,36),(7,42),(8,37),(9,38),(10,39),(11,40),(12,41),(13,30),(14,25),(15,26),(16,27),(17,28),(18,29),(19,46),(20,47),(21,48),(22,43),(23,44),(24,45)]])
90 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 6A | ··· | 6F | 6G | ··· | 6W | 6X | ··· | 6AI | 6AJ | 6AK | 6AL | 6AM | 12A | 12B | 12C | 12D | 12E | ··· | 12L | 12M | ··· | 12R | 12S | ··· | 12Z |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 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 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 6 | 6 | 1 | 1 | 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 | 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 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | - | ||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | C6 | C6 | S3 | D6 | D6 | D6 | C4○D4 | C3×S3 | S3×C6 | S3×C6 | S3×C6 | C3×C4○D4 | D4⋊2S3 | C3×D4⋊2S3 |
| kernel | C6×D4⋊2S3 | C6×Dic6 | S3×C2×C12 | C3×D4⋊2S3 | Dic3×C2×C6 | C6×C3⋊D4 | D4×C3×C6 | C2×D4⋊2S3 | C2×Dic6 | S3×C2×C4 | D4⋊2S3 | C22×Dic3 | C2×C3⋊D4 | C6×D4 | C6×D4 | C2×C12 | C3×D4 | C22×C6 | C3×C6 | C2×D4 | C2×C4 | D4 | C23 | C6 | C6 | C2 |
| # reps | 1 | 1 | 1 | 8 | 2 | 2 | 1 | 2 | 2 | 2 | 16 | 4 | 4 | 2 | 1 | 1 | 4 | 2 | 4 | 2 | 2 | 8 | 4 | 8 | 2 | 4 |
Matrix representation of C6×D4⋊2S3 ►in GL4(𝔽13) generated by
| 10 | 0 | 0 | 0 |
| 0 | 10 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 12 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 12 |
| 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 12 |
| 3 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 12 | 0 | 0 |
| 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 5 |
| 0 | 0 | 8 | 0 |
G:=sub<GL(4,GF(13))| [10,0,0,0,0,10,0,0,0,0,1,0,0,0,0,1],[12,0,0,0,0,12,0,0,0,0,0,1,0,0,12,0],[1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,12],[3,0,0,0,0,9,0,0,0,0,1,0,0,0,0,1],[0,12,0,0,12,0,0,0,0,0,0,8,0,0,5,0] >;
C6×D4⋊2S3 in GAP, Magma, Sage, TeX
C_6\times D_4\rtimes_2S_3
% in TeX
G:=Group("C6xD4:2S3"); // GroupNames label
G:=SmallGroup(288,993);
// by ID
G=gap.SmallGroup(288,993);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-3,268,1571,409,9414]);
// Polycyclic
G:=Group<a,b,c,d,e|a^6=b^4=c^2=d^3=e^2=1,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,e*c*e=b^2*c,e*d*e=d^-1>;
// generators/relations