direct product, metabelian, supersoluble, monomial
Aliases: C3×D8.S3, C24.53D6, C32⋊7SD32, Dic12⋊3C6, C3⋊C16⋊2C6, D8.(C3×S3), C8.5(S3×C6), C6.9(C3×D8), C24.3(C2×C6), (C3×D8).1C6, (C3×D8).7S3, C3⋊2(C3×SD32), C12.4(C3×D4), (C3×C6).31D8, (C3×C12).42D4, C6.31(D4⋊S3), (C3×Dic12)⋊7C2, (C32×D8).1C2, C12.84(C3⋊D4), (C3×C24).14C22, (C3×C3⋊C16)⋊5C2, C2.5(C3×D4⋊S3), C4.2(C3×C3⋊D4), SmallGroup(288,261)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×D8.S3
G = < a,b,c,d,e | a3=b8=c2=d3=1, e2=b4, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe-1=b-1, bd=db, cd=dc, ece-1=b5c, ede-1=d-1 >
Subgroups: 186 in 63 conjugacy classes, 26 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C6, C6, C8, D4, Q8, C32, Dic3, C12, C12, C2×C6, C16, D8, Q16, C3×C6, C3×C6, C24, C24, Dic6, C3×D4, C3×Q8, SD32, C3×Dic3, C3×C12, C62, C3⋊C16, C48, Dic12, C3×D8, C3×D8, C3×Q16, C3×C24, C3×Dic6, D4×C32, D8.S3, C3×SD32, C3×C3⋊C16, C3×Dic12, C32×D8, C3×D8.S3
Quotients: C1, C2, C3, C22, S3, C6, D4, D6, C2×C6, D8, C3×S3, C3⋊D4, C3×D4, SD32, S3×C6, D4⋊S3, C3×D8, C3×C3⋊D4, D8.S3, C3×SD32, C3×D4⋊S3, C3×D8.S3
►On 48 points
(1 33 25)(2 34 26)(3 35 27)(4 36 28)(5 37 29)(6 38 30)(7 39 31)(8 40 32)(9 46 20)(10 47 21)(11 48 22)(12 41 23)(13 42 24)(14 43 17)(15 44 18)(16 45 19)
(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 8)(2 7)(3 6)(4 5)(9 15)(10 14)(11 13)(17 21)(18 20)(22 24)(25 32)(26 31)(27 30)(28 29)(33 40)(34 39)(35 38)(36 37)(42 48)(43 47)(44 46)
(1 25 33)(2 26 34)(3 27 35)(4 28 36)(5 29 37)(6 30 38)(7 31 39)(8 32 40)(9 46 20)(10 47 21)(11 48 22)(12 41 23)(13 42 24)(14 43 17)(15 44 18)(16 45 19)
(1 9 5 13)(2 16 6 12)(3 15 7 11)(4 14 8 10)(17 32 21 28)(18 31 22 27)(19 30 23 26)(20 29 24 25)(33 46 37 42)(34 45 38 41)(35 44 39 48)(36 43 40 47)
G:=sub<Sym(48)| (1,33,25)(2,34,26)(3,35,27)(4,36,28)(5,37,29)(6,38,30)(7,39,31)(8,40,32)(9,46,20)(10,47,21)(11,48,22)(12,41,23)(13,42,24)(14,43,17)(15,44,18)(16,45,19), (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), (1,8)(2,7)(3,6)(4,5)(9,15)(10,14)(11,13)(17,21)(18,20)(22,24)(25,32)(26,31)(27,30)(28,29)(33,40)(34,39)(35,38)(36,37)(42,48)(43,47)(44,46), (1,25,33)(2,26,34)(3,27,35)(4,28,36)(5,29,37)(6,30,38)(7,31,39)(8,32,40)(9,46,20)(10,47,21)(11,48,22)(12,41,23)(13,42,24)(14,43,17)(15,44,18)(16,45,19), (1,9,5,13)(2,16,6,12)(3,15,7,11)(4,14,8,10)(17,32,21,28)(18,31,22,27)(19,30,23,26)(20,29,24,25)(33,46,37,42)(34,45,38,41)(35,44,39,48)(36,43,40,47)>;
G:=Group( (1,33,25)(2,34,26)(3,35,27)(4,36,28)(5,37,29)(6,38,30)(7,39,31)(8,40,32)(9,46,20)(10,47,21)(11,48,22)(12,41,23)(13,42,24)(14,43,17)(15,44,18)(16,45,19), (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), (1,8)(2,7)(3,6)(4,5)(9,15)(10,14)(11,13)(17,21)(18,20)(22,24)(25,32)(26,31)(27,30)(28,29)(33,40)(34,39)(35,38)(36,37)(42,48)(43,47)(44,46), (1,25,33)(2,26,34)(3,27,35)(4,28,36)(5,29,37)(6,30,38)(7,31,39)(8,32,40)(9,46,20)(10,47,21)(11,48,22)(12,41,23)(13,42,24)(14,43,17)(15,44,18)(16,45,19), (1,9,5,13)(2,16,6,12)(3,15,7,11)(4,14,8,10)(17,32,21,28)(18,31,22,27)(19,30,23,26)(20,29,24,25)(33,46,37,42)(34,45,38,41)(35,44,39,48)(36,43,40,47) );
G=PermutationGroup([[(1,33,25),(2,34,26),(3,35,27),(4,36,28),(5,37,29),(6,38,30),(7,39,31),(8,40,32),(9,46,20),(10,47,21),(11,48,22),(12,41,23),(13,42,24),(14,43,17),(15,44,18),(16,45,19)], [(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)], [(1,8),(2,7),(3,6),(4,5),(9,15),(10,14),(11,13),(17,21),(18,20),(22,24),(25,32),(26,31),(27,30),(28,29),(33,40),(34,39),(35,38),(36,37),(42,48),(43,47),(44,46)], [(1,25,33),(2,26,34),(3,27,35),(4,28,36),(5,29,37),(6,30,38),(7,31,39),(8,32,40),(9,46,20),(10,47,21),(11,48,22),(12,41,23),(13,42,24),(14,43,17),(15,44,18),(16,45,19)], [(1,9,5,13),(2,16,6,12),(3,15,7,11),(4,14,8,10),(17,32,21,28),(18,31,22,27),(19,30,23,26),(20,29,24,25),(33,46,37,42),(34,45,38,41),(35,44,39,48),(36,43,40,47)]])
54 conjugacy classes
| class | 1 | 2A | 2B | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 6A | 6B | 6C | 6D | 6E | 6F | ··· | 6M | 8A | 8B | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 16A | 16B | 16C | 16D | 24A | 24B | 24C | 24D | 24E | ··· | 24J | 48A | ··· | 48H |
| order | 1 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | ··· | 6 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 16 | 16 | 16 | 16 | 24 | 24 | 24 | 24 | 24 | ··· | 24 | 48 | ··· | 48 |
| size | 1 | 1 | 8 | 1 | 1 | 2 | 2 | 2 | 2 | 24 | 1 | 1 | 2 | 2 | 2 | 8 | ··· | 8 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 24 | 24 | 6 | 6 | 6 | 6 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 6 | ··· | 6 |
54 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | ||||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | S3 | D4 | D6 | D8 | C3×S3 | C3⋊D4 | C3×D4 | SD32 | S3×C6 | C3×D8 | C3×C3⋊D4 | C3×SD32 | D4⋊S3 | D8.S3 | C3×D4⋊S3 | C3×D8.S3 |
| kernel | C3×D8.S3 | C3×C3⋊C16 | C3×Dic12 | C32×D8 | D8.S3 | C3⋊C16 | Dic12 | C3×D8 | C3×D8 | C3×C12 | C24 | C3×C6 | D8 | C12 | C12 | C32 | C8 | C6 | C4 | C3 | C6 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 2 | 4 | 4 | 8 | 1 | 2 | 2 | 4 |
Matrix representation of C3×D8.S3 ►in GL4(𝔽7) generated by
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 3 | 3 | 0 | 2 |
| 1 | 3 | 1 | 1 |
| 2 | 2 | 6 | 3 |
| 3 | 4 | 2 | 3 |
| 0 | 6 | 4 | 3 |
| 4 | 4 | 4 | 4 |
| 3 | 0 | 0 | 4 |
| 0 | 6 | 6 | 3 |
| 4 | 4 | 4 | 6 |
| 1 | 5 | 6 | 3 |
| 5 | 5 | 6 | 1 |
| 3 | 4 | 5 | 4 |
| 6 | 3 | 3 | 1 |
| 6 | 3 | 6 | 6 |
| 5 | 4 | 6 | 4 |
| 0 | 3 | 2 | 6 |
G:=sub<GL(4,GF(7))| [4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[3,1,2,3,3,3,2,4,0,1,6,2,2,1,3,3],[0,4,3,0,6,4,0,6,4,4,0,6,3,4,4,3],[4,1,5,3,4,5,5,4,4,6,6,5,6,3,1,4],[6,6,5,0,3,3,4,3,3,6,6,2,1,6,4,6] >;
C3×D8.S3 in GAP, Magma, Sage, TeX
C_3\times D_8.S_3
% in TeX
G:=Group("C3xD8.S3"); // GroupNames label
G:=SmallGroup(288,261);
// by ID
G=gap.SmallGroup(288,261);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-3,336,197,1011,514,192,2524,1271,102,9414]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^8=c^2=d^3=1,e^2=b^4,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=e*b*e^-1=b^-1,b*d=d*b,c*d=d*c,e*c*e^-1=b^5*c,e*d*e^-1=d^-1>;
// generators/relations