direct product, metabelian, supersoluble, monomial
Aliases: C6×D4⋊S3, C62.121D4, C3⋊3(C6×D8), C6⋊2(C3×D8), (C3×C6)⋊6D8, (C6×D4)⋊1C6, (C6×D4)⋊8S3, D4⋊3(S3×C6), D12⋊5(C2×C6), (C3×D4)⋊21D6, (C2×D12)⋊8C6, C6.44(C6×D4), C32⋊13(C2×D8), (C6×D12)⋊10C2, C12.14(C3×D4), (C3×C12).84D4, (C2×C12).323D6, (C3×D12)⋊23C22, C12.87(C3⋊D4), C12.11(C22×C6), (C3×C12).82C23, C12.162(C22×S3), (C6×C12).117C22, (D4×C32)⋊14C22, (C2×C3⋊C8)⋊4C6, C3⋊C8⋊7(C2×C6), (D4×C3×C6)⋊1C2, (C6×C3⋊C8)⋊13C2, C4.11(S3×C2×C6), (C3×D4)⋊3(C2×C6), (C2×D4)⋊1(C3×S3), C4.5(C3×C3⋊D4), C2.8(C6×C3⋊D4), (C3×C3⋊C8)⋊32C22, (C2×C4).47(S3×C6), (C2×C6).47(C3×D4), (C2×C12).28(C2×C6), (C3×C6).254(C2×D4), C6.145(C2×C3⋊D4), C22.21(C3×C3⋊D4), (C2×C6).114(C3⋊D4), SmallGroup(288,702)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C6×D4⋊S3
G = < a,b,c,d,e | a6=b4=c2=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe=b-1, bd=db, cd=dc, ece=bc, ede=d-1 >
Subgroups: 490 in 179 conjugacy classes, 66 normal (34 characteristic)
C1, C2, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C6, C8, C2×C4, D4, D4, C23, C32, C12, C12, D6, C2×C6, C2×C6, C2×C8, D8, C2×D4, C2×D4, C3×S3, C3×C6, C3×C6, C3×C6, C3⋊C8, C24, D12, D12, C2×C12, C2×C12, C3×D4, C3×D4, C22×S3, C22×C6, C2×D8, C3×C12, S3×C6, C62, C62, C2×C3⋊C8, D4⋊S3, C2×C24, C3×D8, C2×D12, C6×D4, C6×D4, C3×C3⋊C8, C3×D12, C3×D12, C6×C12, D4×C32, D4×C32, S3×C2×C6, C2×C62, C2×D4⋊S3, C6×D8, C6×C3⋊C8, C3×D4⋊S3, C6×D12, D4×C3×C6, C6×D4⋊S3
Quotients: C1, C2, C3, C22, S3, C6, D4, C23, D6, C2×C6, D8, C2×D4, C3×S3, C3⋊D4, C3×D4, C22×S3, C22×C6, C2×D8, S3×C6, D4⋊S3, C3×D8, C2×C3⋊D4, C6×D4, C3×C3⋊D4, S3×C2×C6, C2×D4⋊S3, C6×D8, C3×D4⋊S3, C6×C3⋊D4, C6×D4⋊S3
►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 11 19 29)(2 12 20 30)(3 7 21 25)(4 8 22 26)(5 9 23 27)(6 10 24 28)(13 47 33 37)(14 48 34 38)(15 43 35 39)(16 44 36 40)(17 45 31 41)(18 46 32 42)
(1 29)(2 30)(3 25)(4 26)(5 27)(6 28)(7 21)(8 22)(9 23)(10 24)(11 19)(12 20)(37 47)(38 48)(39 43)(40 44)(41 45)(42 46)
(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 42)(2 37)(3 38)(4 39)(5 40)(6 41)(7 34)(8 35)(9 36)(10 31)(11 32)(12 33)(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), (1,11,19,29)(2,12,20,30)(3,7,21,25)(4,8,22,26)(5,9,23,27)(6,10,24,28)(13,47,33,37)(14,48,34,38)(15,43,35,39)(16,44,36,40)(17,45,31,41)(18,46,32,42), (1,29)(2,30)(3,25)(4,26)(5,27)(6,28)(7,21)(8,22)(9,23)(10,24)(11,19)(12,20)(37,47)(38,48)(39,43)(40,44)(41,45)(42,46), (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), (1,42)(2,37)(3,38)(4,39)(5,40)(6,41)(7,34)(8,35)(9,36)(10,31)(11,32)(12,33)(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), (1,11,19,29)(2,12,20,30)(3,7,21,25)(4,8,22,26)(5,9,23,27)(6,10,24,28)(13,47,33,37)(14,48,34,38)(15,43,35,39)(16,44,36,40)(17,45,31,41)(18,46,32,42), (1,29)(2,30)(3,25)(4,26)(5,27)(6,28)(7,21)(8,22)(9,23)(10,24)(11,19)(12,20)(37,47)(38,48)(39,43)(40,44)(41,45)(42,46), (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), (1,42)(2,37)(3,38)(4,39)(5,40)(6,41)(7,34)(8,35)(9,36)(10,31)(11,32)(12,33)(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)], [(1,11,19,29),(2,12,20,30),(3,7,21,25),(4,8,22,26),(5,9,23,27),(6,10,24,28),(13,47,33,37),(14,48,34,38),(15,43,35,39),(16,44,36,40),(17,45,31,41),(18,46,32,42)], [(1,29),(2,30),(3,25),(4,26),(5,27),(6,28),(7,21),(8,22),(9,23),(10,24),(11,19),(12,20),(37,47),(38,48),(39,43),(40,44),(41,45),(42,46)], [(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)], [(1,42),(2,37),(3,38),(4,39),(5,40),(6,41),(7,34),(8,35),(9,36),(10,31),(11,32),(12,33),(13,30),(14,25),(15,26),(16,27),(17,28),(18,29),(19,46),(20,47),(21,48),(22,43),(23,44),(24,45)]])
72 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 6A | ··· | 6F | 6G | ··· | 6O | 6P | ··· | 6AE | 6AF | 6AG | 6AH | 6AI | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | ··· | 12J | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 12 | 12 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 12 | 12 | 12 | 12 | 6 | 6 | 6 | 6 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 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 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | ||||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | S3 | D4 | D4 | D6 | D6 | D8 | C3×S3 | C3⋊D4 | C3×D4 | C3⋊D4 | C3×D4 | S3×C6 | S3×C6 | C3×D8 | C3×C3⋊D4 | C3×C3⋊D4 | D4⋊S3 | C3×D4⋊S3 |
| kernel | C6×D4⋊S3 | C6×C3⋊C8 | C3×D4⋊S3 | C6×D12 | D4×C3×C6 | C2×D4⋊S3 | C2×C3⋊C8 | D4⋊S3 | C2×D12 | C6×D4 | C6×D4 | C3×C12 | C62 | C2×C12 | C3×D4 | C3×C6 | C2×D4 | C12 | C12 | C2×C6 | C2×C6 | C2×C4 | D4 | C6 | C4 | C22 | C6 | C2 |
| # reps | 1 | 1 | 4 | 1 | 1 | 2 | 2 | 8 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 4 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 8 | 4 | 4 | 2 | 4 |
Matrix representation of C6×D4⋊S3 ►in GL4(𝔽73) generated by
| 9 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 72 |
| 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 |
| 8 | 0 | 0 | 0 |
| 0 | 64 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 72 | 0 | 0 |
| 72 | 0 | 0 | 0 |
| 0 | 0 | 57 | 16 |
| 0 | 0 | 16 | 16 |
G:=sub<GL(4,GF(73))| [9,0,0,0,0,9,0,0,0,0,72,0,0,0,0,72],[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],[8,0,0,0,0,64,0,0,0,0,1,0,0,0,0,1],[0,72,0,0,72,0,0,0,0,0,57,16,0,0,16,16] >;
C6×D4⋊S3 in GAP, Magma, Sage, TeX
C_6\times D_4\rtimes S_3
% in TeX
G:=Group("C6xD4:S3"); // GroupNames label
G:=SmallGroup(288,702);
// by ID
G=gap.SmallGroup(288,702);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,590,2524,648,102,9414]);
// 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=e*b*e=b^-1,b*d=d*b,c*d=d*c,e*c*e=b*c,e*d*e=d^-1>;
// generators/relations