direct product, metabelian, supersoluble, monomial
Aliases: S3×C8⋊S3, C24⋊20D6, C8⋊6S32, C3⋊C8⋊26D6, (S3×C8)⋊7S3, D6.4(C4×S3), (S3×C24)⋊13C2, C24⋊S3⋊7C2, (C4×S3).27D6, C3⋊1(S3×M4(2)), (C3×C24)⋊20C22, (C3×S3)⋊1M4(2), D6.Dic3⋊9C2, Dic3.7(C4×S3), (S3×Dic3).2C4, C6.D6.2C4, C32⋊1(C2×M4(2)), C12.31D6⋊9C2, (S3×C12).1C22, C12.135(C22×S3), (C3×C12).136C23, C32⋊4C8⋊14C22, C2.4(C4×S32), C6.2(S3×C2×C4), (S3×C3⋊C8)⋊10C2, (C4×S32).1C2, (C2×S32).3C4, C4.82(C2×S32), C3⋊1(C2×C8⋊S3), (C3×C8⋊S3)⋊9C2, (C3×C3⋊C8)⋊22C22, (S3×C6).9(C2×C4), (C3×C6).2(C22×C4), (C4×C3⋊S3).57C22, C3⋊Dic3.17(C2×C4), (C3×Dic3).1(C2×C4), (C2×C3⋊S3).13(C2×C4), SmallGroup(288,438)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for S3×C8⋊S3
G = < a,b,c,d,e | a3=b2=c8=d3=e2=1, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece=c5, ede=d-1 >
Subgroups: 498 in 146 conjugacy classes, 54 normal (50 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, S3, C6, C6, C8, C8, C2×C4, C23, C32, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C2×C8, M4(2), C22×C4, C3×S3, C3×S3, C3⋊S3, C3×C6, C3⋊C8, C3⋊C8, C24, C24, C4×S3, C4×S3, C2×Dic3, C2×C12, C22×S3, C2×M4(2), C3×Dic3, C3⋊Dic3, C3×C12, S32, S3×C6, C2×C3⋊S3, S3×C8, S3×C8, C8⋊S3, C8⋊S3, C2×C3⋊C8, C4.Dic3, C2×C24, C3×M4(2), S3×C2×C4, C3×C3⋊C8, C32⋊4C8, C3×C24, S3×Dic3, C6.D6, S3×C12, C4×C3⋊S3, C2×S32, C2×C8⋊S3, S3×M4(2), S3×C3⋊C8, D6.Dic3, C12.31D6, S3×C24, C3×C8⋊S3, C24⋊S3, C4×S32, S3×C8⋊S3
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D6, M4(2), C22×C4, C4×S3, C22×S3, C2×M4(2), S32, C8⋊S3, S3×C2×C4, C2×S32, C2×C8⋊S3, S3×M4(2), C4×S32, S3×C8⋊S3
►On 48 points
(1 30 40)(2 31 33)(3 32 34)(4 25 35)(5 26 36)(6 27 37)(7 28 38)(8 29 39)(9 48 23)(10 41 24)(11 42 17)(12 43 18)(13 44 19)(14 45 20)(15 46 21)(16 47 22)
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 9)(8 10)(17 30)(18 31)(19 32)(20 25)(21 26)(22 27)(23 28)(24 29)(33 43)(34 44)(35 45)(36 46)(37 47)(38 48)(39 41)(40 42)
(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 40)(2 31 33)(3 32 34)(4 25 35)(5 26 36)(6 27 37)(7 28 38)(8 29 39)(9 23 48)(10 24 41)(11 17 42)(12 18 43)(13 19 44)(14 20 45)(15 21 46)(16 22 47)
(1 11)(2 16)(3 13)(4 10)(5 15)(6 12)(7 9)(8 14)(17 40)(18 37)(19 34)(20 39)(21 36)(22 33)(23 38)(24 35)(25 41)(26 46)(27 43)(28 48)(29 45)(30 42)(31 47)(32 44)
G:=sub<Sym(48)| (1,30,40)(2,31,33)(3,32,34)(4,25,35)(5,26,36)(6,27,37)(7,28,38)(8,29,39)(9,48,23)(10,41,24)(11,42,17)(12,43,18)(13,44,19)(14,45,20)(15,46,21)(16,47,22), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,9)(8,10)(17,30)(18,31)(19,32)(20,25)(21,26)(22,27)(23,28)(24,29)(33,43)(34,44)(35,45)(36,46)(37,47)(38,48)(39,41)(40,42), (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), (1,30,40)(2,31,33)(3,32,34)(4,25,35)(5,26,36)(6,27,37)(7,28,38)(8,29,39)(9,23,48)(10,24,41)(11,17,42)(12,18,43)(13,19,44)(14,20,45)(15,21,46)(16,22,47), (1,11)(2,16)(3,13)(4,10)(5,15)(6,12)(7,9)(8,14)(17,40)(18,37)(19,34)(20,39)(21,36)(22,33)(23,38)(24,35)(25,41)(26,46)(27,43)(28,48)(29,45)(30,42)(31,47)(32,44)>;
G:=Group( (1,30,40)(2,31,33)(3,32,34)(4,25,35)(5,26,36)(6,27,37)(7,28,38)(8,29,39)(9,48,23)(10,41,24)(11,42,17)(12,43,18)(13,44,19)(14,45,20)(15,46,21)(16,47,22), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,9)(8,10)(17,30)(18,31)(19,32)(20,25)(21,26)(22,27)(23,28)(24,29)(33,43)(34,44)(35,45)(36,46)(37,47)(38,48)(39,41)(40,42), (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), (1,30,40)(2,31,33)(3,32,34)(4,25,35)(5,26,36)(6,27,37)(7,28,38)(8,29,39)(9,23,48)(10,24,41)(11,17,42)(12,18,43)(13,19,44)(14,20,45)(15,21,46)(16,22,47), (1,11)(2,16)(3,13)(4,10)(5,15)(6,12)(7,9)(8,14)(17,40)(18,37)(19,34)(20,39)(21,36)(22,33)(23,38)(24,35)(25,41)(26,46)(27,43)(28,48)(29,45)(30,42)(31,47)(32,44) );
G=PermutationGroup([[(1,30,40),(2,31,33),(3,32,34),(4,25,35),(5,26,36),(6,27,37),(7,28,38),(8,29,39),(9,48,23),(10,41,24),(11,42,17),(12,43,18),(13,44,19),(14,45,20),(15,46,21),(16,47,22)], [(1,11),(2,12),(3,13),(4,14),(5,15),(6,16),(7,9),(8,10),(17,30),(18,31),(19,32),(20,25),(21,26),(22,27),(23,28),(24,29),(33,43),(34,44),(35,45),(36,46),(37,47),(38,48),(39,41),(40,42)], [(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)], [(1,30,40),(2,31,33),(3,32,34),(4,25,35),(5,26,36),(6,27,37),(7,28,38),(8,29,39),(9,23,48),(10,24,41),(11,17,42),(12,18,43),(13,19,44),(14,20,45),(15,21,46),(16,22,47)], [(1,11),(2,16),(3,13),(4,10),(5,15),(6,12),(7,9),(8,14),(17,40),(18,37),(19,34),(20,39),(21,36),(22,33),(23,38),(24,35),(25,41),(26,46),(27,43),(28,48),(29,45),(30,42),(31,47),(32,44)]])
54 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 3C | 4A | 4B | 4C | 4D | 4E | 4F | 6A | 6B | 6C | 6D | 6E | 6F | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 12I | 24A | 24B | 24C | 24D | 24E | ··· | 24J | 24K | 24L | 24M | 24N | 24O | 24P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | 24 | 24 | 24 | 24 | ··· | 24 | 24 | 24 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 3 | 3 | 6 | 18 | 2 | 2 | 4 | 1 | 1 | 3 | 3 | 6 | 18 | 2 | 2 | 4 | 6 | 6 | 12 | 2 | 2 | 6 | 6 | 6 | 6 | 18 | 18 | 2 | 2 | 2 | 2 | 4 | 4 | 6 | 6 | 12 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | 12 | 12 |
54 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | S3 | S3 | D6 | D6 | D6 | M4(2) | C4×S3 | C4×S3 | C8⋊S3 | S32 | C2×S32 | S3×M4(2) | C4×S32 | S3×C8⋊S3 |
| kernel | S3×C8⋊S3 | S3×C3⋊C8 | D6.Dic3 | C12.31D6 | S3×C24 | C3×C8⋊S3 | C24⋊S3 | C4×S32 | S3×Dic3 | C6.D6 | C2×S32 | S3×C8 | C8⋊S3 | C3⋊C8 | C24 | C4×S3 | C3×S3 | Dic3 | D6 | S3 | C8 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | 1 | 1 | 2 | 2 | 4 |
Matrix representation of S3×C8⋊S3 ►in GL4(𝔽5) generated by
| 4 | 4 | 0 | 3 |
| 2 | 0 | 4 | 0 |
| 0 | 4 | 4 | 1 |
| 2 | 0 | 3 | 0 |
| 0 | 4 | 0 | 3 |
| 3 | 0 | 1 | 0 |
| 0 | 4 | 0 | 1 |
| 3 | 0 | 2 | 0 |
| 1 | 0 | 2 | 0 |
| 0 | 1 | 0 | 2 |
| 1 | 0 | 4 | 0 |
| 0 | 1 | 0 | 4 |
| 3 | 0 | 2 | 0 |
| 0 | 3 | 0 | 2 |
| 1 | 0 | 1 | 0 |
| 0 | 1 | 0 | 1 |
| 4 | 0 | 2 | 0 |
| 0 | 1 | 0 | 3 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 4 |
G:=sub<GL(4,GF(5))| [4,2,0,2,4,0,4,0,0,4,4,3,3,0,1,0],[0,3,0,3,4,0,4,0,0,1,0,2,3,0,1,0],[1,0,1,0,0,1,0,1,2,0,4,0,0,2,0,4],[3,0,1,0,0,3,0,1,2,0,1,0,0,2,0,1],[4,0,0,0,0,1,0,0,2,0,1,0,0,3,0,4] >;
S3×C8⋊S3 in GAP, Magma, Sage, TeX
S_3\times C_8\rtimes S_3
% in TeX
G:=Group("S3xC8:S3"); // GroupNames label
G:=SmallGroup(288,438);
// by ID
G=gap.SmallGroup(288,438);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,219,58,80,1356,9414]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^2=c^8=d^3=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,c*d=d*c,e*c*e=c^5,e*d*e=d^-1>;
// generators/relations