direct product, metabelian, supersoluble, monomial
Aliases: S3×Q8⋊3S3, D12⋊14D6, Dic6⋊15D6, Q8⋊6S32, (C4×S3)⋊9D6, (S3×Q8)⋊9S3, (C3×Q8)⋊11D6, (S3×D12)⋊11C2, (S3×C12)⋊9C22, C6.25(S3×C23), (C3×C6).25C24, D12⋊S3⋊12C2, D6.6D6⋊12C2, C12.26D6⋊7C2, C3⋊D12⋊6C22, (C3×D12)⋊16C22, (S3×C6).13C23, (C3×C12).37C23, C12.37(C22×S3), (S3×Dic3)⋊8C22, D6.13(C22×S3), C12⋊S3⋊12C22, (C3×Dic6)⋊17C22, C6.D6⋊10C22, C3⋊Dic3.41C23, (Q8×C32)⋊10C22, Dic3.14(C22×S3), (C3×Dic3).26C23, (C4×S32)⋊7C2, C4.37(C2×S32), C3⋊5(S3×C4○D4), (C3×S3×Q8)⋊10C2, C32⋊9(C2×C4○D4), (C4×C3⋊S3)⋊6C22, C2.27(C22×S32), C3⋊3(C2×Q8⋊3S3), (C3×S3)⋊3(C4○D4), (C2×S32).13C22, (C3×Q8⋊3S3)⋊8C2, (C2×C3⋊S3).28C23, SmallGroup(288,966)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for S3×Q8⋊3S3
G = < a,b,c,d,e,f | a3=b2=c4=e3=f2=1, d2=c2, bab=a-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, dcd-1=fcf=c-1, ce=ec, de=ed, df=fd, fef=e-1 >
Subgroups: 1306 in 348 conjugacy classes, 112 normal (24 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, S3, C6, C6, C2×C4, D4, Q8, Q8, C23, C32, Dic3, Dic3, Dic3, C12, C12, D6, D6, D6, C2×C6, C22×C4, C2×D4, C2×Q8, C4○D4, C3×S3, C3×S3, C3⋊S3, C3×C6, Dic6, C4×S3, C4×S3, D12, D12, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C3×Q8, C3×Q8, C22×S3, C2×C4○D4, C3×Dic3, C3×Dic3, C3⋊Dic3, C3×C12, S32, S3×C6, S3×C6, C2×C3⋊S3, S3×C2×C4, C2×D12, C4○D12, S3×D4, D4⋊2S3, S3×Q8, Q8⋊3S3, Q8⋊3S3, C6×Q8, C3×C4○D4, S3×Dic3, S3×Dic3, C6.D6, C3⋊D12, C3×Dic6, S3×C12, C3×D12, C4×C3⋊S3, C12⋊S3, Q8×C32, C2×S32, C2×Q8⋊3S3, S3×C4○D4, D12⋊S3, D6.6D6, C4×S32, S3×D12, C3×S3×Q8, C3×Q8⋊3S3, C12.26D6, S3×Q8⋊3S3
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C24, C22×S3, C2×C4○D4, S32, Q8⋊3S3, S3×C23, C2×S32, C2×Q8⋊3S3, S3×C4○D4, C22×S32, S3×Q8⋊3S3
►On 48 points
(1 19 14)(2 20 15)(3 17 16)(4 18 13)(5 10 47)(6 11 48)(7 12 45)(8 9 46)(21 32 27)(22 29 28)(23 30 25)(24 31 26)(33 37 42)(34 38 43)(35 39 44)(36 40 41)
(1 33)(2 34)(3 35)(4 36)(5 28)(6 25)(7 26)(8 27)(9 32)(10 29)(11 30)(12 31)(13 40)(14 37)(15 38)(16 39)(17 44)(18 41)(19 42)(20 43)(21 46)(22 47)(23 48)(24 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 24 3 22)(2 23 4 21)(5 37 7 39)(6 40 8 38)(9 43 11 41)(10 42 12 44)(13 27 15 25)(14 26 16 28)(17 29 19 31)(18 32 20 30)(33 45 35 47)(34 48 36 46)
(1 19 14)(2 20 15)(3 17 16)(4 18 13)(5 47 10)(6 48 11)(7 45 12)(8 46 9)(21 32 27)(22 29 28)(23 30 25)(24 31 26)(33 42 37)(34 43 38)(35 44 39)(36 41 40)
(1 34)(2 33)(3 36)(4 35)(5 32)(6 31)(7 30)(8 29)(9 28)(10 27)(11 26)(12 25)(13 44)(14 43)(15 42)(16 41)(17 40)(18 39)(19 38)(20 37)(21 47)(22 46)(23 45)(24 48)
G:=sub<Sym(48)| (1,19,14)(2,20,15)(3,17,16)(4,18,13)(5,10,47)(6,11,48)(7,12,45)(8,9,46)(21,32,27)(22,29,28)(23,30,25)(24,31,26)(33,37,42)(34,38,43)(35,39,44)(36,40,41), (1,33)(2,34)(3,35)(4,36)(5,28)(6,25)(7,26)(8,27)(9,32)(10,29)(11,30)(12,31)(13,40)(14,37)(15,38)(16,39)(17,44)(18,41)(19,42)(20,43)(21,46)(22,47)(23,48)(24,45), (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), (1,24,3,22)(2,23,4,21)(5,37,7,39)(6,40,8,38)(9,43,11,41)(10,42,12,44)(13,27,15,25)(14,26,16,28)(17,29,19,31)(18,32,20,30)(33,45,35,47)(34,48,36,46), (1,19,14)(2,20,15)(3,17,16)(4,18,13)(5,47,10)(6,48,11)(7,45,12)(8,46,9)(21,32,27)(22,29,28)(23,30,25)(24,31,26)(33,42,37)(34,43,38)(35,44,39)(36,41,40), (1,34)(2,33)(3,36)(4,35)(5,32)(6,31)(7,30)(8,29)(9,28)(10,27)(11,26)(12,25)(13,44)(14,43)(15,42)(16,41)(17,40)(18,39)(19,38)(20,37)(21,47)(22,46)(23,45)(24,48)>;
G:=Group( (1,19,14)(2,20,15)(3,17,16)(4,18,13)(5,10,47)(6,11,48)(7,12,45)(8,9,46)(21,32,27)(22,29,28)(23,30,25)(24,31,26)(33,37,42)(34,38,43)(35,39,44)(36,40,41), (1,33)(2,34)(3,35)(4,36)(5,28)(6,25)(7,26)(8,27)(9,32)(10,29)(11,30)(12,31)(13,40)(14,37)(15,38)(16,39)(17,44)(18,41)(19,42)(20,43)(21,46)(22,47)(23,48)(24,45), (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), (1,24,3,22)(2,23,4,21)(5,37,7,39)(6,40,8,38)(9,43,11,41)(10,42,12,44)(13,27,15,25)(14,26,16,28)(17,29,19,31)(18,32,20,30)(33,45,35,47)(34,48,36,46), (1,19,14)(2,20,15)(3,17,16)(4,18,13)(5,47,10)(6,48,11)(7,45,12)(8,46,9)(21,32,27)(22,29,28)(23,30,25)(24,31,26)(33,42,37)(34,43,38)(35,44,39)(36,41,40), (1,34)(2,33)(3,36)(4,35)(5,32)(6,31)(7,30)(8,29)(9,28)(10,27)(11,26)(12,25)(13,44)(14,43)(15,42)(16,41)(17,40)(18,39)(19,38)(20,37)(21,47)(22,46)(23,45)(24,48) );
G=PermutationGroup([[(1,19,14),(2,20,15),(3,17,16),(4,18,13),(5,10,47),(6,11,48),(7,12,45),(8,9,46),(21,32,27),(22,29,28),(23,30,25),(24,31,26),(33,37,42),(34,38,43),(35,39,44),(36,40,41)], [(1,33),(2,34),(3,35),(4,36),(5,28),(6,25),(7,26),(8,27),(9,32),(10,29),(11,30),(12,31),(13,40),(14,37),(15,38),(16,39),(17,44),(18,41),(19,42),(20,43),(21,46),(22,47),(23,48),(24,45)], [(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)], [(1,24,3,22),(2,23,4,21),(5,37,7,39),(6,40,8,38),(9,43,11,41),(10,42,12,44),(13,27,15,25),(14,26,16,28),(17,29,19,31),(18,32,20,30),(33,45,35,47),(34,48,36,46)], [(1,19,14),(2,20,15),(3,17,16),(4,18,13),(5,47,10),(6,48,11),(7,45,12),(8,46,9),(21,32,27),(22,29,28),(23,30,25),(24,31,26),(33,42,37),(34,43,38),(35,44,39),(36,41,40)], [(1,34),(2,33),(3,36),(4,35),(5,32),(6,31),(7,30),(8,29),(9,28),(10,27),(11,26),(12,25),(13,44),(14,43),(15,42),(16,41),(17,40),(18,39),(19,38),(20,37),(21,47),(22,46),(23,45),(24,48)]])
45 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 3A | 3B | 3C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 12A | ··· | 12F | 12G | 12H | 12I | 12J | 12K | 12L | 12M | 12N |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 12 | ··· | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 3 | 3 | 6 | 6 | 6 | 18 | 18 | 18 | 2 | 2 | 4 | 2 | 2 | 2 | 3 | 3 | 6 | 6 | 6 | 9 | 9 | 2 | 2 | 4 | 6 | 6 | 12 | 12 | 12 | 4 | ··· | 4 | 6 | 6 | 8 | 8 | 8 | 12 | 12 | 12 |
45 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | S3 | D6 | D6 | D6 | D6 | C4○D4 | S32 | Q8⋊3S3 | C2×S32 | S3×C4○D4 | S3×Q8⋊3S3 |
| kernel | S3×Q8⋊3S3 | D12⋊S3 | D6.6D6 | C4×S32 | S3×D12 | C3×S3×Q8 | C3×Q8⋊3S3 | C12.26D6 | S3×Q8 | Q8⋊3S3 | Dic6 | C4×S3 | D12 | C3×Q8 | C3×S3 | Q8 | S3 | C4 | C3 | C1 |
| # reps | 1 | 3 | 3 | 3 | 3 | 1 | 1 | 1 | 1 | 1 | 3 | 6 | 3 | 2 | 4 | 1 | 2 | 3 | 2 | 1 |
Matrix representation of S3×Q8⋊3S3 ►in GL6(𝔽13)
| 0 | 1 | 0 | 0 | 0 | 0 |
| 12 | 12 | 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 | 1 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 8 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 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 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(6,GF(13))| [0,12,0,0,0,0,1,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,1,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,8,0,0,0,0,8,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[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],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;
S3×Q8⋊3S3 in GAP, Magma, Sage, TeX
S_3\times Q_8\rtimes_3S_3
% in TeX
G:=Group("S3xQ8:3S3"); // GroupNames label
G:=SmallGroup(288,966);
// by ID
G=gap.SmallGroup(288,966);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,100,346,185,80,1356,9414]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^3=b^2=c^4=e^3=f^2=1,d^2=c^2,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,d*c*d^-1=f*c*f=c^-1,c*e=e*c,d*e=e*d,d*f=f*d,f*e*f=e^-1>;
// generators/relations