direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: S3×C4○D8, D8⋊14D6, Q16⋊12D6, SD16⋊14D6, D24⋊18C22, C12.14C24, C24.45C23, D12.9C23, Dic6.9C23, Dic12⋊16C22, (S3×D8)⋊8C2, (C2×C8)⋊27D6, C4○D24⋊6C2, C4○D4⋊10D6, (S3×Q16)⋊8C2, D8⋊3S3⋊8C2, (C2×C24)⋊4C22, (S3×SD16)⋊7C2, (C4×S3).54D4, D6.66(C2×D4), C4.221(S3×D4), C3⋊C8.24C23, D24⋊C2⋊8C2, C22.4(S3×D4), (S3×C8)⋊16C22, D4⋊S3⋊12C22, Q8.13D6⋊1C2, Q8.7D6⋊7C2, C12.380(C2×D4), C4○D12⋊5C22, (C3×D8)⋊12C22, C8.42(C22×S3), C4.14(S3×C23), D4.8(C22×S3), (S3×D4).6C22, (C3×D4).8C23, C24⋊C2⋊20C22, (C3×Q8).8C23, (S3×Q8).5C22, D4⋊2S3⋊8C22, (C4×S3).29C23, D4.S3⋊11C22, Dic3.71(C2×D4), (C3×Q16)⋊10C22, Q8⋊3S3⋊8C22, C3⋊Q16⋊10C22, (C22×S3).64D4, C6.115(C22×D4), Q8.18(C22×S3), (C2×C12).531C23, (C2×Dic3).124D4, Q8⋊2S3⋊11C22, (C3×SD16)⋊15C22, (S3×C2×C8)⋊1C2, C3⋊5(C2×C4○D8), C2.88(C2×S3×D4), (S3×C4○D4)⋊1C2, (C3×C4○D8)⋊2C2, (C2×C3⋊C8)⋊37C22, (C2×C6).11(C2×D4), (C3×C4○D4)⋊1C22, (S3×C2×C4).261C22, (C2×C4).618(C22×S3), SmallGroup(192,1326)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for S3×C4○D8
G = < a,b,c,d,e | a3=b2=c4=e2=1, d4=c2, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=c2d3 >
Subgroups: 712 in 266 conjugacy classes, 101 normal (51 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, S3, C6, C6, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C2×C8, C2×C8, D8, D8, SD16, SD16, Q16, Q16, C22×C4, C2×D4, C2×Q8, C4○D4, C4○D4, C3⋊C8, C24, Dic6, Dic6, C4×S3, C4×S3, D12, D12, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C22×S3, C22×S3, C22×C8, C2×D8, C2×SD16, C2×Q16, C4○D8, C4○D8, C2×C4○D4, S3×C8, C24⋊C2, D24, Dic12, C2×C3⋊C8, D4⋊S3, D4.S3, Q8⋊2S3, C3⋊Q16, C2×C24, C3×D8, C3×SD16, C3×Q16, S3×C2×C4, S3×C2×C4, C4○D12, C4○D12, S3×D4, S3×D4, D4⋊2S3, D4⋊2S3, S3×Q8, Q8⋊3S3, C3×C4○D4, C2×C4○D8, S3×C2×C8, C4○D24, S3×D8, D8⋊3S3, S3×SD16, Q8.7D6, S3×Q16, D24⋊C2, Q8.13D6, C3×C4○D8, S3×C4○D4, S3×C4○D8
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C24, C22×S3, C4○D8, C22×D4, S3×D4, S3×C23, C2×C4○D8, C2×S3×D4, S3×C4○D8
►On 48 points
(1 40 27)(2 33 28)(3 34 29)(4 35 30)(5 36 31)(6 37 32)(7 38 25)(8 39 26)(9 24 45)(10 17 46)(11 18 47)(12 19 48)(13 20 41)(14 21 42)(15 22 43)(16 23 44)
(9 45)(10 46)(11 47)(12 48)(13 41)(14 42)(15 43)(16 44)(25 38)(26 39)(27 40)(28 33)(29 34)(30 35)(31 36)(32 37)
(1 19 5 23)(2 20 6 24)(3 21 7 17)(4 22 8 18)(9 28 13 32)(10 29 14 25)(11 30 15 26)(12 31 16 27)(33 41 37 45)(34 42 38 46)(35 43 39 47)(36 44 40 48)
(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 8)(3 7)(4 6)(9 15)(10 14)(11 13)(17 21)(18 20)(22 24)(25 29)(26 28)(30 32)(33 39)(34 38)(35 37)(41 47)(42 46)(43 45)
G:=sub<Sym(48)| (1,40,27)(2,33,28)(3,34,29)(4,35,30)(5,36,31)(6,37,32)(7,38,25)(8,39,26)(9,24,45)(10,17,46)(11,18,47)(12,19,48)(13,20,41)(14,21,42)(15,22,43)(16,23,44), (9,45)(10,46)(11,47)(12,48)(13,41)(14,42)(15,43)(16,44)(25,38)(26,39)(27,40)(28,33)(29,34)(30,35)(31,36)(32,37), (1,19,5,23)(2,20,6,24)(3,21,7,17)(4,22,8,18)(9,28,13,32)(10,29,14,25)(11,30,15,26)(12,31,16,27)(33,41,37,45)(34,42,38,46)(35,43,39,47)(36,44,40,48), (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), (2,8)(3,7)(4,6)(9,15)(10,14)(11,13)(17,21)(18,20)(22,24)(25,29)(26,28)(30,32)(33,39)(34,38)(35,37)(41,47)(42,46)(43,45)>;
G:=Group( (1,40,27)(2,33,28)(3,34,29)(4,35,30)(5,36,31)(6,37,32)(7,38,25)(8,39,26)(9,24,45)(10,17,46)(11,18,47)(12,19,48)(13,20,41)(14,21,42)(15,22,43)(16,23,44), (9,45)(10,46)(11,47)(12,48)(13,41)(14,42)(15,43)(16,44)(25,38)(26,39)(27,40)(28,33)(29,34)(30,35)(31,36)(32,37), (1,19,5,23)(2,20,6,24)(3,21,7,17)(4,22,8,18)(9,28,13,32)(10,29,14,25)(11,30,15,26)(12,31,16,27)(33,41,37,45)(34,42,38,46)(35,43,39,47)(36,44,40,48), (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), (2,8)(3,7)(4,6)(9,15)(10,14)(11,13)(17,21)(18,20)(22,24)(25,29)(26,28)(30,32)(33,39)(34,38)(35,37)(41,47)(42,46)(43,45) );
G=PermutationGroup([[(1,40,27),(2,33,28),(3,34,29),(4,35,30),(5,36,31),(6,37,32),(7,38,25),(8,39,26),(9,24,45),(10,17,46),(11,18,47),(12,19,48),(13,20,41),(14,21,42),(15,22,43),(16,23,44)], [(9,45),(10,46),(11,47),(12,48),(13,41),(14,42),(15,43),(16,44),(25,38),(26,39),(27,40),(28,33),(29,34),(30,35),(31,36),(32,37)], [(1,19,5,23),(2,20,6,24),(3,21,7,17),(4,22,8,18),(9,28,13,32),(10,29,14,25),(11,30,15,26),(12,31,16,27),(33,41,37,45),(34,42,38,46),(35,43,39,47),(36,44,40,48)], [(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)], [(2,8),(3,7),(4,6),(9,15),(10,14),(11,13),(17,21),(18,20),(22,24),(25,29),(26,28),(30,32),(33,39),(34,38),(35,37),(41,47),(42,46),(43,45)]])
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 6A | 6B | 6C | 6D | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 12A | 12B | 12C | 12D | 12E | 24A | 24B | 24C | 24D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 2 | 3 | 3 | 4 | 4 | 6 | 12 | 12 | 2 | 1 | 1 | 2 | 3 | 3 | 4 | 4 | 6 | 12 | 12 | 2 | 4 | 8 | 8 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 2 | 2 | 4 | 8 | 8 | 4 | 4 | 4 | 4 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D4 | D6 | D6 | D6 | D6 | D6 | C4○D8 | S3×D4 | S3×D4 | S3×C4○D8 |
| kernel | S3×C4○D8 | S3×C2×C8 | C4○D24 | S3×D8 | D8⋊3S3 | S3×SD16 | Q8.7D6 | S3×Q16 | D24⋊C2 | Q8.13D6 | C3×C4○D8 | S3×C4○D4 | C4○D8 | C4×S3 | C2×Dic3 | C22×S3 | C2×C8 | D8 | SD16 | Q16 | C4○D4 | S3 | C4 | C22 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 1 | 2 | 1 | 2 | 1 | 1 | 1 | 1 | 2 | 1 | 2 | 8 | 1 | 1 | 4 |
Matrix representation of S3×C4○D8 ►in GL4(𝔽73) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 72 | 72 |
| 0 | 0 | 1 | 0 |
| 72 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 72 | 72 |
| 27 | 0 | 0 | 0 |
| 0 | 27 | 0 | 0 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 72 |
| 16 | 57 | 0 | 0 |
| 16 | 16 | 0 | 0 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 72 |
| 1 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,72,1,0,0,72,0],[72,0,0,0,0,72,0,0,0,0,1,72,0,0,0,72],[27,0,0,0,0,27,0,0,0,0,72,0,0,0,0,72],[16,16,0,0,57,16,0,0,0,0,72,0,0,0,0,72],[1,0,0,0,0,72,0,0,0,0,1,0,0,0,0,1] >;
S3×C4○D8 in GAP, Magma, Sage, TeX
S_3\times C_4\circ D_8
% in TeX
G:=Group("S3xC4oD8"); // GroupNames label
G:=SmallGroup(192,1326);
// by ID
G=gap.SmallGroup(192,1326);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,387,570,185,438,235,102,6278]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^2=c^4=e^2=1,d^4=c^2,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,c*e=e*c,e*d*e=c^2*d^3>;
// generators/relations