direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: S3×C4⋊D4, C4⋊5(S3×D4), C4⋊C4⋊20D6, D6⋊3(C2×D4), C12⋊5(C2×D4), (C4×S3)⋊11D4, (C2×D4)⋊20D6, C22⋊2(S3×D4), C22⋊C4⋊25D6, Dic3⋊7(C2×D4), (C22×C4)⋊41D6, Dic3⋊D4⋊18C2, D6⋊3D4⋊16C2, C12⋊D4⋊20C2, C12⋊7D4⋊32C2, D6⋊C4⋊25C22, (C22×S3)⋊11D4, (C6×D4)⋊10C22, C6.63(C22×D4), D6.37(C4○D4), (C2×D12)⋊22C22, (C2×C6).148C24, C4⋊Dic3⋊29C22, C23.14D6⋊11C2, (C2×C12).593C23, Dic3⋊C4⋊14C22, (C22×C12)⋊19C22, C23.24(C22×S3), C6.D4⋊20C22, (S3×C23).45C22, C22.169(S3×C23), (C22×C6).186C23, (C2×Dic3).69C23, (C22×S3).183C23, (C22×Dic3)⋊44C22, (C2×S3×D4)⋊8C2, (C2×C6)⋊2(C2×D4), C3⋊3(C2×C4⋊D4), (S3×C4⋊C4)⋊20C2, C2.36(C2×S3×D4), (S3×C22×C4)⋊3C2, (C3×C4⋊C4)⋊7C22, (S3×C22⋊C4)⋊4C2, (S3×C2×C4)⋊56C22, C2.37(S3×C4○D4), (C3×C4⋊D4)⋊10C2, C6.150(C2×C4○D4), (C2×C3⋊D4)⋊12C22, (C3×C22⋊C4)⋊9C22, (C2×C4).37(C22×S3), SmallGroup(192,1163)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for S3×C4⋊D4
G = < a,b,c,d,e | a3=b2=c4=d4=e2=1, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=ece=c-1, ede=d-1 >
Subgroups: 1264 in 426 conjugacy classes, 121 normal (43 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, S3, S3, C6, C6, C2×C4, C2×C4, C2×C4, D4, C23, C23, C23, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C2×C6, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C24, C4×S3, C4×S3, D12, C2×Dic3, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C2×C12, C3×D4, C22×S3, C22×S3, C22×S3, C22×C6, C22×C6, C2×C22⋊C4, C2×C4⋊C4, C4⋊D4, C4⋊D4, C23×C4, C22×D4, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C6.D4, C3×C22⋊C4, C3×C4⋊C4, S3×C2×C4, S3×C2×C4, S3×C2×C4, C2×D12, C2×D12, S3×D4, C22×Dic3, C2×C3⋊D4, C22×C12, C6×D4, C6×D4, S3×C23, S3×C23, C2×C4⋊D4, S3×C22⋊C4, Dic3⋊D4, S3×C4⋊C4, C12⋊D4, C12⋊7D4, D6⋊3D4, C23.14D6, C3×C4⋊D4, S3×C22×C4, C2×S3×D4, C2×S3×D4, S3×C4⋊D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C24, C22×S3, C4⋊D4, C22×D4, C2×C4○D4, S3×D4, S3×C23, C2×C4⋊D4, C2×S3×D4, S3×C4○D4, S3×C4⋊D4
►On 48 points
(1 27 9)(2 28 10)(3 25 11)(4 26 12)(5 46 13)(6 47 14)(7 48 15)(8 45 16)(17 33 30)(18 34 31)(19 35 32)(20 36 29)(21 41 39)(22 42 40)(23 43 37)(24 44 38)
(9 27)(10 28)(11 25)(12 26)(13 46)(14 47)(15 48)(16 45)(17 33)(18 34)(19 35)(20 36)(21 41)(22 42)(23 43)(24 44)
(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 39 5)(2 29 40 8)(3 32 37 7)(4 31 38 6)(9 33 41 13)(10 36 42 16)(11 35 43 15)(12 34 44 14)(17 21 46 27)(18 24 47 26)(19 23 48 25)(20 22 45 28)
(1 3)(5 32)(6 31)(7 30)(8 29)(9 11)(13 35)(14 34)(15 33)(16 36)(17 48)(18 47)(19 46)(20 45)(21 23)(25 27)(37 39)(41 43)
G:=sub<Sym(48)| (1,27,9)(2,28,10)(3,25,11)(4,26,12)(5,46,13)(6,47,14)(7,48,15)(8,45,16)(17,33,30)(18,34,31)(19,35,32)(20,36,29)(21,41,39)(22,42,40)(23,43,37)(24,44,38), (9,27)(10,28)(11,25)(12,26)(13,46)(14,47)(15,48)(16,45)(17,33)(18,34)(19,35)(20,36)(21,41)(22,42)(23,43)(24,44), (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), (1,30,39,5)(2,29,40,8)(3,32,37,7)(4,31,38,6)(9,33,41,13)(10,36,42,16)(11,35,43,15)(12,34,44,14)(17,21,46,27)(18,24,47,26)(19,23,48,25)(20,22,45,28), (1,3)(5,32)(6,31)(7,30)(8,29)(9,11)(13,35)(14,34)(15,33)(16,36)(17,48)(18,47)(19,46)(20,45)(21,23)(25,27)(37,39)(41,43)>;
G:=Group( (1,27,9)(2,28,10)(3,25,11)(4,26,12)(5,46,13)(6,47,14)(7,48,15)(8,45,16)(17,33,30)(18,34,31)(19,35,32)(20,36,29)(21,41,39)(22,42,40)(23,43,37)(24,44,38), (9,27)(10,28)(11,25)(12,26)(13,46)(14,47)(15,48)(16,45)(17,33)(18,34)(19,35)(20,36)(21,41)(22,42)(23,43)(24,44), (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), (1,30,39,5)(2,29,40,8)(3,32,37,7)(4,31,38,6)(9,33,41,13)(10,36,42,16)(11,35,43,15)(12,34,44,14)(17,21,46,27)(18,24,47,26)(19,23,48,25)(20,22,45,28), (1,3)(5,32)(6,31)(7,30)(8,29)(9,11)(13,35)(14,34)(15,33)(16,36)(17,48)(18,47)(19,46)(20,45)(21,23)(25,27)(37,39)(41,43) );
G=PermutationGroup([[(1,27,9),(2,28,10),(3,25,11),(4,26,12),(5,46,13),(6,47,14),(7,48,15),(8,45,16),(17,33,30),(18,34,31),(19,35,32),(20,36,29),(21,41,39),(22,42,40),(23,43,37),(24,44,38)], [(9,27),(10,28),(11,25),(12,26),(13,46),(14,47),(15,48),(16,45),(17,33),(18,34),(19,35),(20,36),(21,41),(22,42),(23,43),(24,44)], [(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)], [(1,30,39,5),(2,29,40,8),(3,32,37,7),(4,31,38,6),(9,33,41,13),(10,36,42,16),(11,35,43,15),(12,34,44,14),(17,21,46,27),(18,24,47,26),(19,23,48,25),(20,22,45,28)], [(1,3),(5,32),(6,31),(7,30),(8,29),(9,11),(13,35),(14,34),(15,33),(16,36),(17,48),(18,47),(19,46),(20,45),(21,23),(25,27),(37,39),(41,43)]])
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 2L | 2M | 2N | 2O | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 12A | 12B | 12C | 12D | 12E | 12F |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 6 | 6 | 12 | 12 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 6 | 6 | 6 | 6 | 12 | 12 | 2 | 2 | 2 | 4 | 4 | 8 | 8 | 4 | 4 | 4 | 4 | 8 | 8 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | D6 | D6 | C4○D4 | S3×D4 | S3×D4 | S3×C4○D4 |
| kernel | S3×C4⋊D4 | S3×C22⋊C4 | Dic3⋊D4 | S3×C4⋊C4 | C12⋊D4 | C12⋊7D4 | D6⋊3D4 | C23.14D6 | C3×C4⋊D4 | S3×C22×C4 | C2×S3×D4 | C4⋊D4 | C4×S3 | C22×S3 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | D6 | C4 | C22 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 3 | 1 | 4 | 4 | 2 | 1 | 1 | 3 | 4 | 2 | 2 | 2 |
Matrix representation of S3×C4⋊D4 ►in GL6(𝔽13)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 0 | 0 | 0 | 1 | 12 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 5 | 0 | 0 |
| 0 | 0 | 10 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 5 | 0 | 0 | 0 | 0 |
| 5 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 3 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 10 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,12,12],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[0,1,0,0,0,0,12,0,0,0,0,0,0,0,12,10,0,0,0,0,5,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[0,5,0,0,0,0,5,0,0,0,0,0,0,0,1,3,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,1,0,0,0,0,0,0,12,10,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
S3×C4⋊D4 in GAP, Magma, Sage, TeX
S_3\times C_4\rtimes D_4
% in TeX
G:=Group("S3xC4:D4"); // GroupNames label
G:=SmallGroup(192,1163);
// by ID
G=gap.SmallGroup(192,1163);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,100,794,297,6278]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^2=c^4=d^4=e^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d^-1=e*c*e=c^-1,e*d*e=d^-1>;
// generators/relations