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, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C8, C2×C4, C2×C4, C23, Dic3, C12, D6, D6, C2×C6, C2×C8, C2×C8, M4(2), C22×C4, C3⋊C8, C24, C4×S3, C2×Dic3, C2×C12, C22×S3, C2×M4(2), C8⋊S3, C2×C3⋊C8, C2×C24, S3×C2×C4, C2×C8⋊S3
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D6, M4(2), C22×C4, C4×S3, C22×S3, C2×M4(2), C8⋊S3, S3×C2×C4, C2×C8⋊S3
►On 48 points
(1 22)(2 23)(3 24)(4 17)(5 18)(6 19)(7 20)(8 21)(9 34)(10 35)(11 36)(12 37)(13 38)(14 39)(15 40)(16 33)(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 9)(2 46 10)(3 47 11)(4 48 12)(5 41 13)(6 42 14)(7 43 15)(8 44 16)(17 26 37)(18 27 38)(19 28 39)(20 29 40)(21 30 33)(22 31 34)(23 32 35)(24 25 36)
(2 6)(4 8)(9 45)(10 42)(11 47)(12 44)(13 41)(14 46)(15 43)(16 48)(17 21)(19 23)(25 36)(26 33)(27 38)(28 35)(29 40)(30 37)(31 34)(32 39)
G:=sub<Sym(48)| (1,22)(2,23)(3,24)(4,17)(5,18)(6,19)(7,20)(8,21)(9,34)(10,35)(11,36)(12,37)(13,38)(14,39)(15,40)(16,33)(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), (1,45,9)(2,46,10)(3,47,11)(4,48,12)(5,41,13)(6,42,14)(7,43,15)(8,44,16)(17,26,37)(18,27,38)(19,28,39)(20,29,40)(21,30,33)(22,31,34)(23,32,35)(24,25,36), (2,6)(4,8)(9,45)(10,42)(11,47)(12,44)(13,41)(14,46)(15,43)(16,48)(17,21)(19,23)(25,36)(26,33)(27,38)(28,35)(29,40)(30,37)(31,34)(32,39)>;
G:=Group( (1,22)(2,23)(3,24)(4,17)(5,18)(6,19)(7,20)(8,21)(9,34)(10,35)(11,36)(12,37)(13,38)(14,39)(15,40)(16,33)(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), (1,45,9)(2,46,10)(3,47,11)(4,48,12)(5,41,13)(6,42,14)(7,43,15)(8,44,16)(17,26,37)(18,27,38)(19,28,39)(20,29,40)(21,30,33)(22,31,34)(23,32,35)(24,25,36), (2,6)(4,8)(9,45)(10,42)(11,47)(12,44)(13,41)(14,46)(15,43)(16,48)(17,21)(19,23)(25,36)(26,33)(27,38)(28,35)(29,40)(30,37)(31,34)(32,39) );
G=PermutationGroup([[(1,22),(2,23),(3,24),(4,17),(5,18),(6,19),(7,20),(8,21),(9,34),(10,35),(11,36),(12,37),(13,38),(14,39),(15,40),(16,33),(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)], [(1,45,9),(2,46,10),(3,47,11),(4,48,12),(5,41,13),(6,42,14),(7,43,15),(8,44,16),(17,26,37),(18,27,38),(19,28,39),(20,29,40),(21,30,33),(22,31,34),(23,32,35),(24,25,36)], [(2,6),(4,8),(9,45),(10,42),(11,47),(12,44),(13,41),(14,46),(15,43),(16,48),(17,21),(19,23),(25,36),(26,33),(27,38),(28,35),(29,40),(30,37),(31,34),(32,39)]])
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