direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: S3×M5(2), C16⋊6D6, C48⋊7C22, C24.74C23, (S3×C16)⋊7C2, (C4×S3).1C8, (S3×C8).4C4, C8.35(C4×S3), C4.15(S3×C8), D6.C8⋊5C2, D6.6(C2×C8), C3⋊2(C2×M5(2)), C3⋊C16⋊11C22, C12.12(C2×C8), C24.44(C2×C4), (C2×C8).272D6, C22.7(S3×C8), (C3×M5(2))⋊5C2, (C22×S3).4C8, C8.60(C22×S3), C6.15(C22×C8), Dic3.7(C2×C8), (C2×Dic3).6C8, C12.C8⋊13C2, (S3×C8).18C22, (C2×C24).273C22, C12.131(C22×C4), (C2×C3⋊C8).8C4, (S3×C2×C4).8C4, C2.16(S3×C2×C8), C3⋊C8.22(C2×C4), (S3×C2×C8).18C2, C4.105(S3×C2×C4), (C2×C6).5(C2×C8), (C4×S3).34(C2×C4), (C2×C12).72(C2×C4), (C2×C4).148(C4×S3), SmallGroup(192,465)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for S3×M5(2)
G = < a,b,c,d | a3=b2=c16=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c9 >
Subgroups: 168 in 90 conjugacy classes, 53 normal (41 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, S3, C6, C6, C8, C8, C2×C4, C2×C4, C23, Dic3, C12, D6, D6, C2×C6, C16, C16, C2×C8, C2×C8, C22×C4, C3⋊C8, C24, C4×S3, C2×Dic3, C2×C12, C22×S3, C2×C16, M5(2), M5(2), C22×C8, C3⋊C16, C48, S3×C8, C2×C3⋊C8, C2×C24, S3×C2×C4, C2×M5(2), S3×C16, D6.C8, C12.C8, C3×M5(2), S3×C2×C8, S3×M5(2)
Quotients: C1, C2, C4, C22, S3, C8, C2×C4, C23, D6, C2×C8, C22×C4, C4×S3, C22×S3, M5(2), C22×C8, S3×C8, S3×C2×C4, C2×M5(2), S3×C2×C8, S3×M5(2)
►On 48 points
(1 41 30)(2 42 31)(3 43 32)(4 44 17)(5 45 18)(6 46 19)(7 47 20)(8 48 21)(9 33 22)(10 34 23)(11 35 24)(12 36 25)(13 37 26)(14 38 27)(15 39 28)(16 40 29)
(1 9)(2 10)(3 11)(4 12)(5 13)(6 14)(7 15)(8 16)(17 36)(18 37)(19 38)(20 39)(21 40)(22 41)(23 42)(24 43)(25 44)(26 45)(27 46)(28 47)(29 48)(30 33)(31 34)(32 35)
(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)
(2 10)(4 12)(6 14)(8 16)(17 25)(19 27)(21 29)(23 31)(34 42)(36 44)(38 46)(40 48)
G:=sub<Sym(48)| (1,41,30)(2,42,31)(3,43,32)(4,44,17)(5,45,18)(6,46,19)(7,47,20)(8,48,21)(9,33,22)(10,34,23)(11,35,24)(12,36,25)(13,37,26)(14,38,27)(15,39,28)(16,40,29), (1,9)(2,10)(3,11)(4,12)(5,13)(6,14)(7,15)(8,16)(17,36)(18,37)(19,38)(20,39)(21,40)(22,41)(23,42)(24,43)(25,44)(26,45)(27,46)(28,47)(29,48)(30,33)(31,34)(32,35), (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), (2,10)(4,12)(6,14)(8,16)(17,25)(19,27)(21,29)(23,31)(34,42)(36,44)(38,46)(40,48)>;
G:=Group( (1,41,30)(2,42,31)(3,43,32)(4,44,17)(5,45,18)(6,46,19)(7,47,20)(8,48,21)(9,33,22)(10,34,23)(11,35,24)(12,36,25)(13,37,26)(14,38,27)(15,39,28)(16,40,29), (1,9)(2,10)(3,11)(4,12)(5,13)(6,14)(7,15)(8,16)(17,36)(18,37)(19,38)(20,39)(21,40)(22,41)(23,42)(24,43)(25,44)(26,45)(27,46)(28,47)(29,48)(30,33)(31,34)(32,35), (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), (2,10)(4,12)(6,14)(8,16)(17,25)(19,27)(21,29)(23,31)(34,42)(36,44)(38,46)(40,48) );
G=PermutationGroup([[(1,41,30),(2,42,31),(3,43,32),(4,44,17),(5,45,18),(6,46,19),(7,47,20),(8,48,21),(9,33,22),(10,34,23),(11,35,24),(12,36,25),(13,37,26),(14,38,27),(15,39,28),(16,40,29)], [(1,9),(2,10),(3,11),(4,12),(5,13),(6,14),(7,15),(8,16),(17,36),(18,37),(19,38),(20,39),(21,40),(22,41),(23,42),(24,43),(25,44),(26,45),(27,46),(28,47),(29,48),(30,33),(31,34),(32,35)], [(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)], [(2,10),(4,12),(6,14),(8,16),(17,25),(19,27),(21,29),(23,31),(34,42),(36,44),(38,46),(40,48)]])
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 6A | 6B | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | 8J | 8K | 8L | 12A | 12B | 12C | 16A | ··· | 16H | 16I | ··· | 16P | 24A | 24B | 24C | 24D | 24E | 24F | 48A | ··· | 48H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 16 | ··· | 16 | 16 | ··· | 16 | 24 | 24 | 24 | 24 | 24 | 24 | 48 | ··· | 48 |
| size | 1 | 1 | 2 | 3 | 3 | 6 | 2 | 1 | 1 | 2 | 3 | 3 | 6 | 2 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 3 | 3 | 3 | 3 | 6 | 6 | 2 | 2 | 4 | 2 | ··· | 2 | 6 | ··· | 6 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | ||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C8 | C8 | C8 | S3 | D6 | D6 | C4×S3 | C4×S3 | M5(2) | S3×C8 | S3×C8 | S3×M5(2) |
| kernel | S3×M5(2) | S3×C16 | D6.C8 | C12.C8 | C3×M5(2) | S3×C2×C8 | S3×C8 | C2×C3⋊C8 | S3×C2×C4 | C4×S3 | C2×Dic3 | C22×S3 | M5(2) | C16 | C2×C8 | C8 | C2×C4 | S3 | C4 | C22 | C1 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 4 | 2 | 2 | 8 | 4 | 4 | 1 | 2 | 1 | 2 | 2 | 8 | 4 | 4 | 4 |
Matrix representation of S3×M5(2) ►in GL4(𝔽97) generated by
| 0 | 96 | 0 | 0 |
| 1 | 96 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 96 | 0 | 0 |
| 0 | 96 | 0 | 0 |
| 0 | 0 | 96 | 0 |
| 0 | 0 | 0 | 96 |
| 64 | 0 | 0 | 0 |
| 0 | 64 | 0 | 0 |
| 0 | 0 | 47 | 3 |
| 0 | 0 | 24 | 50 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 96 |
G:=sub<GL(4,GF(97))| [0,1,0,0,96,96,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,96,96,0,0,0,0,96,0,0,0,0,96],[64,0,0,0,0,64,0,0,0,0,47,24,0,0,3,50],[1,0,0,0,0,1,0,0,0,0,1,1,0,0,0,96] >;
S3×M5(2) in GAP, Magma, Sage, TeX
S_3\times M_5(2)
% in TeX
G:=Group("S3xM5(2)"); // GroupNames label
G:=SmallGroup(192,465);
// by ID
G=gap.SmallGroup(192,465);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,387,58,80,102,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^2=c^16=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^9>;
// generators/relations