direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C8⋊S3, C8⋊9D6, C6⋊1M4(2), C24⋊12C22, C12.36C23, (C2×C8)⋊6S3, C4○(C8⋊S3), (C2×C24)⋊11C2, (C4×S3).3C4, C4.24(C4×S3), C3⋊C8⋊10C22, D6.5(C2×C4), (C2×C4).98D6, C3⋊1(C2×M4(2)), C12.27(C2×C4), (C22×S3).3C4, C4.36(C22×S3), C6.13(C22×C4), C22.14(C4×S3), (C2×Dic3).5C4, Dic3.6(C2×C4), (C4×S3).14C22, (C2×C12).111C22, (C2×C3⋊C8)⋊11C2, C2.14(S3×C2×C4), (S3×C2×C4).10C2, (C2×C6).15(C2×C4), SmallGroup(96,107)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C8⋊S3
G = < a,b,c,d | a2=b8=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b5, dcd=c-1 >
Subgroups: 130 in 68 conjugacy classes, 41 normal (19 characteristic)
C1, C2, C2 [×2], C2 [×2], C3, C4 [×2], C4 [×2], C22, C22 [×4], S3 [×2], C6, C6 [×2], C8 [×2], C8 [×2], C2×C4, C2×C4 [×5], C23, Dic3 [×2], C12 [×2], D6 [×2], D6 [×2], C2×C6, C2×C8, C2×C8, M4(2) [×4], C22×C4, C3⋊C8 [×2], C24 [×2], C4×S3 [×4], C2×Dic3, C2×C12, C22×S3, C2×M4(2), C8⋊S3 [×4], C2×C3⋊C8, C2×C24, S3×C2×C4, C2×C8⋊S3
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], C23, D6 [×3], M4(2) [×2], C22×C4, C4×S3 [×2], C22×S3, C2×M4(2), C8⋊S3 [×2], S3×C2×C4, C2×C8⋊S3
►On 48 points
(1 17)(2 18)(3 19)(4 20)(5 21)(6 22)(7 23)(8 24)(9 35)(10 36)(11 37)(12 38)(13 39)(14 40)(15 33)(16 34)(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 45 13)(2 46 14)(3 47 15)(4 48 16)(5 41 9)(6 42 10)(7 43 11)(8 44 12)(17 31 39)(18 32 40)(19 25 33)(20 26 34)(21 27 35)(22 28 36)(23 29 37)(24 30 38)
(2 6)(4 8)(9 41)(10 46)(11 43)(12 48)(13 45)(14 42)(15 47)(16 44)(18 22)(20 24)(25 33)(26 38)(27 35)(28 40)(29 37)(30 34)(31 39)(32 36)
G:=sub<Sym(48)| (1,17)(2,18)(3,19)(4,20)(5,21)(6,22)(7,23)(8,24)(9,35)(10,36)(11,37)(12,38)(13,39)(14,40)(15,33)(16,34)(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), (1,45,13)(2,46,14)(3,47,15)(4,48,16)(5,41,9)(6,42,10)(7,43,11)(8,44,12)(17,31,39)(18,32,40)(19,25,33)(20,26,34)(21,27,35)(22,28,36)(23,29,37)(24,30,38), (2,6)(4,8)(9,41)(10,46)(11,43)(12,48)(13,45)(14,42)(15,47)(16,44)(18,22)(20,24)(25,33)(26,38)(27,35)(28,40)(29,37)(30,34)(31,39)(32,36)>;
G:=Group( (1,17)(2,18)(3,19)(4,20)(5,21)(6,22)(7,23)(8,24)(9,35)(10,36)(11,37)(12,38)(13,39)(14,40)(15,33)(16,34)(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), (1,45,13)(2,46,14)(3,47,15)(4,48,16)(5,41,9)(6,42,10)(7,43,11)(8,44,12)(17,31,39)(18,32,40)(19,25,33)(20,26,34)(21,27,35)(22,28,36)(23,29,37)(24,30,38), (2,6)(4,8)(9,41)(10,46)(11,43)(12,48)(13,45)(14,42)(15,47)(16,44)(18,22)(20,24)(25,33)(26,38)(27,35)(28,40)(29,37)(30,34)(31,39)(32,36) );
G=PermutationGroup([(1,17),(2,18),(3,19),(4,20),(5,21),(6,22),(7,23),(8,24),(9,35),(10,36),(11,37),(12,38),(13,39),(14,40),(15,33),(16,34),(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)], [(1,45,13),(2,46,14),(3,47,15),(4,48,16),(5,41,9),(6,42,10),(7,43,11),(8,44,12),(17,31,39),(18,32,40),(19,25,33),(20,26,34),(21,27,35),(22,28,36),(23,29,37),(24,30,38)], [(2,6),(4,8),(9,41),(10,46),(11,43),(12,48),(13,45),(14,42),(15,47),(16,44),(18,22),(20,24),(25,33),(26,38),(27,35),(28,40),(29,37),(30,34),(31,39),(32,36)])
C2×C8⋊S3 is a maximal subgroup of
C8.25D12 C8⋊6D12 D6.C42 C8⋊9D12 Dic3⋊5M4(2) D6.4C42 D6⋊M4(2) D6⋊C8⋊C2 Dic3⋊M4(2) C3⋊C8⋊26D4 C4⋊C4⋊19D6 D4⋊(C4×S3) C3⋊C8⋊1D4 C3⋊C8⋊D4 (S3×Q8)⋊C4 Q8⋊7(C4×S3) C3⋊(C8⋊D4) C3⋊C8.D4 D6⋊3M4(2) C12⋊M4(2) C12⋊2M4(2) C42.30D6 C8⋊(C4×S3) C24⋊7D4 C8.2D12 C8⋊S3⋊C4 C8⋊3D12 M4(2).25D6 C24⋊33D4 C24⋊21D4 C24⋊12D4 C24⋊8D4 C24.36D4 C2×S3×M4(2) M4(2)⋊28D6 SD16⋊D6 C5⋊C8⋊D6
C2×C8⋊S3 is a maximal quotient of
C24⋊12Q8 C42.282D6 C8⋊6D12 Dic3.M4(2) D6⋊M4(2) C3⋊C8⋊26D4 C42.198D6 C42.202D6 C12⋊M4(2) C12⋊2M4(2) C24⋊33D4 C5⋊C8⋊D6
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 12A | 12B | 12C | 12D | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 2 | 1 | 1 | 1 | 1 | 6 | 6 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 2 | 2 | 2 | 2 | 2 | ··· | 2 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | S3 | D6 | D6 | M4(2) | C4×S3 | C4×S3 | C8⋊S3 |
| kernel | C2×C8⋊S3 | C8⋊S3 | C2×C3⋊C8 | C2×C24 | S3×C2×C4 | C4×S3 | C2×Dic3 | C22×S3 | C2×C8 | C8 | C2×C4 | C6 | C4 | C22 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 4 | 2 | 2 | 1 | 2 | 1 | 4 | 2 | 2 | 8 |
Matrix representation of C2×C8⋊S3 ►in GL3(𝔽73) generated by
| 72 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 72 | 0 | 0 |
| 0 | 65 | 16 |
| 0 | 57 | 8 |
| 1 | 0 | 0 |
| 0 | 72 | 1 |
| 0 | 72 | 0 |
| 72 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 1 | 0 |
G:=sub<GL(3,GF(73))| [72,0,0,0,1,0,0,0,1],[72,0,0,0,65,57,0,16,8],[1,0,0,0,72,72,0,1,0],[72,0,0,0,0,1,0,1,0] >;
C2×C8⋊S3 in GAP, Magma, Sage, TeX
C_2\times C_8\rtimes S_3
% in TeX
G:=Group("C2xC8:S3"); // GroupNames label
G:=SmallGroup(96,107);
// by ID
G=gap.SmallGroup(96,107);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-3,362,50,69,2309]);
// 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,d*b*d=b^5,d*c*d=c^-1>;
// generators/relations