direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: S3×C2×C8, C24⋊11C22, C12.35C23, C8○(S3×C8), C4○(S3×C8), C8○(C4×S3), C6⋊1(C2×C8), C3⋊1(C22×C8), Dic3○(C2×C8), C8○(C2×Dic3), (C2×C24)⋊10C2, (C4×S3).5C4, C4.23(C4×S3), C3⋊C8⋊13C22, D6.9(C2×C4), (C2×C4).97D6, C12.26(C2×C4), (C22×S3).5C4, C4.35(C22×S3), C6.12(C22×C4), C22.13(C4×S3), (C2×Dic3).8C4, (C4×S3).17C22, Dic3.10(C2×C4), (C2×C12).110C22, C8○2(C2×C3⋊C8), C2.2(S3×C2×C4), (C2×C3⋊C8)⋊13C2, (C2×C8)○2(C3⋊C8), (S3×C2×C4).12C2, (C2×C8)○(C2×Dic3), (C2×C6).14(C2×C4), (C2×C8)○(C2×C3⋊C8), SmallGroup(96,106)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — S3×C2×C8 |
Generators and relations for S3×C2×C8
G = < a,b,c,d | a2=b8=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 130 in 76 conjugacy classes, 49 normal (19 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C8, C2×C4, C2×C4, C23, Dic3, C12, D6, C2×C6, C2×C8, C2×C8, C22×C4, C3⋊C8, C24, C4×S3, C2×Dic3, C2×C12, C22×S3, C22×C8, S3×C8, C2×C3⋊C8, C2×C24, S3×C2×C4, S3×C2×C8
Quotients: C1, C2, C4, C22, S3, C8, C2×C4, C23, D6, C2×C8, C22×C4, C4×S3, C22×S3, C22×C8, S3×C8, S3×C2×C4, S3×C2×C8
►On 48 points
(1 13)(2 14)(3 15)(4 16)(5 9)(6 10)(7 11)(8 12)(17 39)(18 40)(19 33)(20 34)(21 35)(22 36)(23 37)(24 38)(25 47)(26 48)(27 41)(28 42)(29 43)(30 44)(31 45)(32 46)
(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 28 35)(2 29 36)(3 30 37)(4 31 38)(5 32 39)(6 25 40)(7 26 33)(8 27 34)(9 46 17)(10 47 18)(11 48 19)(12 41 20)(13 42 21)(14 43 22)(15 44 23)(16 45 24)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)(17 42)(18 43)(19 44)(20 45)(21 46)(22 47)(23 48)(24 41)(25 36)(26 37)(27 38)(28 39)(29 40)(30 33)(31 34)(32 35)
G:=sub<Sym(48)| (1,13)(2,14)(3,15)(4,16)(5,9)(6,10)(7,11)(8,12)(17,39)(18,40)(19,33)(20,34)(21,35)(22,36)(23,37)(24,38)(25,47)(26,48)(27,41)(28,42)(29,43)(30,44)(31,45)(32,46), (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), (1,28,35)(2,29,36)(3,30,37)(4,31,38)(5,32,39)(6,25,40)(7,26,33)(8,27,34)(9,46,17)(10,47,18)(11,48,19)(12,41,20)(13,42,21)(14,43,22)(15,44,23)(16,45,24), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,42)(18,43)(19,44)(20,45)(21,46)(22,47)(23,48)(24,41)(25,36)(26,37)(27,38)(28,39)(29,40)(30,33)(31,34)(32,35)>;
G:=Group( (1,13)(2,14)(3,15)(4,16)(5,9)(6,10)(7,11)(8,12)(17,39)(18,40)(19,33)(20,34)(21,35)(22,36)(23,37)(24,38)(25,47)(26,48)(27,41)(28,42)(29,43)(30,44)(31,45)(32,46), (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), (1,28,35)(2,29,36)(3,30,37)(4,31,38)(5,32,39)(6,25,40)(7,26,33)(8,27,34)(9,46,17)(10,47,18)(11,48,19)(12,41,20)(13,42,21)(14,43,22)(15,44,23)(16,45,24), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,42)(18,43)(19,44)(20,45)(21,46)(22,47)(23,48)(24,41)(25,36)(26,37)(27,38)(28,39)(29,40)(30,33)(31,34)(32,35) );
G=PermutationGroup([[(1,13),(2,14),(3,15),(4,16),(5,9),(6,10),(7,11),(8,12),(17,39),(18,40),(19,33),(20,34),(21,35),(22,36),(23,37),(24,38),(25,47),(26,48),(27,41),(28,42),(29,43),(30,44),(31,45),(32,46)], [(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)], [(1,28,35),(2,29,36),(3,30,37),(4,31,38),(5,32,39),(6,25,40),(7,26,33),(8,27,34),(9,46,17),(10,47,18),(11,48,19),(12,41,20),(13,42,21),(14,43,22),(15,44,23),(16,45,24)], [(1,5),(2,6),(3,7),(4,8),(9,13),(10,14),(11,15),(12,16),(17,42),(18,43),(19,44),(20,45),(21,46),(22,47),(23,48),(24,41),(25,36),(26,37),(27,38),(28,39),(29,40),(30,33),(31,34),(32,35)]])
S3×C2×C8 is a maximal subgroup of
D6⋊C16 D6.C42 C8⋊9D12 D6.4C42 C3⋊D4⋊C8 D6⋊C8⋊C2 D6⋊2M4(2) D4⋊2S3⋊C4 D6⋊D8 D6⋊SD16 C4⋊C4.150D6 D6⋊2SD16 D6⋊1Q16 D12⋊C8 D6⋊3M4(2) C42.30D6 (S3×C8)⋊C4 C8⋊8D12 C8.27(C4×S3) D6⋊2D8 D6⋊2Q16 C24⋊D4 D6⋊3D8 C24⋊14D4 D6⋊3Q16
S3×C2×C8 is a maximal quotient of
C42.282D6 Dic3.5M4(2) C3⋊D4⋊C8 Dic6⋊C8 C42.200D6 D12⋊C8 D12.4C8 C16.12D6
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | 6B | 6C | 8A | ··· | 8H | 8I | ··· | 8P | 12A | 12B | 12C | 12D | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 8 | ··· | 8 | 8 | ··· | 8 | 12 | 12 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 2 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 2 | 2 | 2 | 1 | ··· | 1 | 3 | ··· | 3 | 2 | 2 | 2 | 2 | 2 | ··· | 2 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C8 | S3 | D6 | D6 | C4×S3 | C4×S3 | S3×C8 |
| kernel | S3×C2×C8 | S3×C8 | C2×C3⋊C8 | C2×C24 | S3×C2×C4 | C4×S3 | C2×Dic3 | C22×S3 | D6 | C2×C8 | C8 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 4 | 2 | 2 | 16 | 1 | 2 | 1 | 2 | 2 | 8 |
Matrix representation of S3×C2×C8 ►in GL3(𝔽73) generated by
| 72 | 0 | 0 |
| 0 | 72 | 0 |
| 0 | 0 | 72 |
| 72 | 0 | 0 |
| 0 | 22 | 0 |
| 0 | 0 | 22 |
| 1 | 0 | 0 |
| 0 | 72 | 72 |
| 0 | 1 | 0 |
| 72 | 0 | 0 |
| 0 | 72 | 0 |
| 0 | 1 | 1 |
G:=sub<GL(3,GF(73))| [72,0,0,0,72,0,0,0,72],[72,0,0,0,22,0,0,0,22],[1,0,0,0,72,1,0,72,0],[72,0,0,0,72,1,0,0,1] >;
S3×C2×C8 in GAP, Magma, Sage, TeX
S_3\times C_2\times C_8
% in TeX
G:=Group("S3xC2xC8"); // GroupNames label
G:=SmallGroup(96,106);
// by ID
G=gap.SmallGroup(96,106);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-3,50,69,2309]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^8=c^3=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations