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), (C2×D4)⋊20D6, (C4×S3)⋊11D4, C22⋊2(S3×D4), C22⋊C4⋊25D6, Dic3⋊7(C2×D4), (C22×C4)⋊41D6, D6⋊3D4⋊16C2, C12⋊D4⋊20C2, Dic3⋊D4⋊18C2, C12⋊7D4⋊32C2, D6⋊C4⋊25C22, (C6×D4)⋊10C22, (C22×S3)⋊11D4, 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×C6).186C23, C22.169(S3×C23), (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
Subgroups: 1264 in 426 conjugacy classes, 121 normal (43 characteristic)
C1, C2 [×3], C2 [×12], C3, C4 [×2], C4 [×8], C22, C22 [×2], C22 [×40], S3 [×4], S3 [×4], C6 [×3], C6 [×4], C2×C4 [×2], C2×C4 [×2], C2×C4 [×22], D4 [×24], C23, C23 [×2], C23 [×24], Dic3 [×2], Dic3 [×3], C12 [×2], C12 [×3], D6 [×8], D6 [×24], C2×C6, C2×C6 [×2], C2×C6 [×8], C22⋊C4 [×2], C22⋊C4 [×6], C4⋊C4, C4⋊C4 [×3], C22×C4, C22×C4 [×11], C2×D4, C2×D4 [×2], C2×D4 [×21], C24 [×3], C4×S3 [×4], C4×S3 [×10], D12 [×6], C2×Dic3 [×2], C2×Dic3 [×2], C2×Dic3 [×2], C3⋊D4 [×12], C2×C12 [×2], C2×C12 [×2], C2×C12 [×2], C3×D4 [×6], C22×S3 [×2], C22×S3 [×6], C22×S3 [×16], C22×C6, C22×C6 [×2], C2×C22⋊C4 [×2], C2×C4⋊C4, C4⋊D4, C4⋊D4 [×7], C23×C4, C22×D4 [×3], Dic3⋊C4 [×2], C4⋊Dic3, D6⋊C4 [×4], C6.D4 [×2], C3×C22⋊C4 [×2], C3×C4⋊C4, S3×C2×C4 [×4], S3×C2×C4 [×2], S3×C2×C4 [×4], C2×D12, C2×D12 [×2], S3×D4 [×12], C22×Dic3, C2×C3⋊D4 [×6], C22×C12, C6×D4, C6×D4 [×2], S3×C23, S3×C23 [×2], C2×C4⋊D4, S3×C22⋊C4 [×2], Dic3⋊D4 [×2], S3×C4⋊C4, C12⋊D4, C12⋊7D4, D6⋊3D4, C23.14D6 [×2], C3×C4⋊D4, S3×C22×C4, C2×S3×D4, C2×S3×D4 [×2], S3×C4⋊D4
Quotients:
C1, C2 [×15], C22 [×35], S3, D4 [×8], C23 [×15], D6 [×7], C2×D4 [×12], C4○D4 [×2], C24, C22×S3 [×7], C4⋊D4 [×4], C22×D4 [×2], C2×C4○D4, S3×D4 [×4], S3×C23, C2×C4⋊D4, C2×S3×D4 [×2], S3×C4○D4, S3×C4⋊D4
Generators and relations
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 >
►On 48 points
(1 27 9)(2 28 10)(3 25 11)(4 26 12)(5 41 30)(6 42 31)(7 43 32)(8 44 29)(13 17 46)(14 18 47)(15 19 48)(16 20 45)(21 35 39)(22 36 40)(23 33 37)(24 34 38)
(5 41)(6 42)(7 43)(8 44)(9 27)(10 28)(11 25)(12 26)(13 46)(14 47)(15 48)(16 45)(21 35)(22 36)(23 33)(24 34)
(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 17)(2 29 40 20)(3 32 37 19)(4 31 38 18)(5 21 46 27)(6 24 47 26)(7 23 48 25)(8 22 45 28)(9 41 35 13)(10 44 36 16)(11 43 33 15)(12 42 34 14)
(1 3)(5 48)(6 47)(7 46)(8 45)(9 11)(13 43)(14 42)(15 41)(16 44)(17 32)(18 31)(19 30)(20 29)(21 23)(25 27)(33 35)(37 39)
G:=sub<Sym(48)| (1,27,9)(2,28,10)(3,25,11)(4,26,12)(5,41,30)(6,42,31)(7,43,32)(8,44,29)(13,17,46)(14,18,47)(15,19,48)(16,20,45)(21,35,39)(22,36,40)(23,33,37)(24,34,38), (5,41)(6,42)(7,43)(8,44)(9,27)(10,28)(11,25)(12,26)(13,46)(14,47)(15,48)(16,45)(21,35)(22,36)(23,33)(24,34), (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), (1,30,39,17)(2,29,40,20)(3,32,37,19)(4,31,38,18)(5,21,46,27)(6,24,47,26)(7,23,48,25)(8,22,45,28)(9,41,35,13)(10,44,36,16)(11,43,33,15)(12,42,34,14), (1,3)(5,48)(6,47)(7,46)(8,45)(9,11)(13,43)(14,42)(15,41)(16,44)(17,32)(18,31)(19,30)(20,29)(21,23)(25,27)(33,35)(37,39)>;
G:=Group( (1,27,9)(2,28,10)(3,25,11)(4,26,12)(5,41,30)(6,42,31)(7,43,32)(8,44,29)(13,17,46)(14,18,47)(15,19,48)(16,20,45)(21,35,39)(22,36,40)(23,33,37)(24,34,38), (5,41)(6,42)(7,43)(8,44)(9,27)(10,28)(11,25)(12,26)(13,46)(14,47)(15,48)(16,45)(21,35)(22,36)(23,33)(24,34), (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), (1,30,39,17)(2,29,40,20)(3,32,37,19)(4,31,38,18)(5,21,46,27)(6,24,47,26)(7,23,48,25)(8,22,45,28)(9,41,35,13)(10,44,36,16)(11,43,33,15)(12,42,34,14), (1,3)(5,48)(6,47)(7,46)(8,45)(9,11)(13,43)(14,42)(15,41)(16,44)(17,32)(18,31)(19,30)(20,29)(21,23)(25,27)(33,35)(37,39) );
G=PermutationGroup([(1,27,9),(2,28,10),(3,25,11),(4,26,12),(5,41,30),(6,42,31),(7,43,32),(8,44,29),(13,17,46),(14,18,47),(15,19,48),(16,20,45),(21,35,39),(22,36,40),(23,33,37),(24,34,38)], [(5,41),(6,42),(7,43),(8,44),(9,27),(10,28),(11,25),(12,26),(13,46),(14,47),(15,48),(16,45),(21,35),(22,36),(23,33),(24,34)], [(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)], [(1,30,39,17),(2,29,40,20),(3,32,37,19),(4,31,38,18),(5,21,46,27),(6,24,47,26),(7,23,48,25),(8,22,45,28),(9,41,35,13),(10,44,36,16),(11,43,33,15),(12,42,34,14)], [(1,3),(5,48),(6,47),(7,46),(8,45),(9,11),(13,43),(14,42),(15,41),(16,44),(17,32),(18,31),(19,30),(20,29),(21,23),(25,27),(33,35),(37,39)])
Matrix representation ►G ⊆ 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] >;
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 |
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