direct product, metabelian, supersoluble, monomial, A-group
Aliases: S3×C3⋊C8, C12.28D6, D6.2Dic3, Dic3.2Dic3, (C3×S3)⋊C8, C3⋊3(S3×C8), C4.13S32, C32⋊3(C2×C8), (S3×C6).1C4, (C4×S3).3S3, C6.16(C4×S3), (S3×C12).4C2, C32⋊4C8⋊6C2, C2.1(S3×Dic3), C6.1(C2×Dic3), (C3×Dic3).2C4, (C3×C12).27C22, C3⋊1(C2×C3⋊C8), (C3×C3⋊C8)⋊5C2, (C3×C6).8(C2×C4), SmallGroup(144,52)
Series: Derived ►Chief ►Lower central ►Upper central
| C32 — S3×C3⋊C8 |
Generators and relations for S3×C3⋊C8
G = < a,b,c,d | a3=b2=c3=d8=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >
Smallest permutation representation of S3×C3⋊C8
►On 48 points
(1 30 35)(2 31 36)(3 32 37)(4 25 38)(5 26 39)(6 27 40)(7 28 33)(8 29 34)(9 20 41)(10 21 42)(11 22 43)(12 23 44)(13 24 45)(14 17 46)(15 18 47)(16 19 48)
(1 10)(2 11)(3 12)(4 13)(5 14)(6 15)(7 16)(8 9)(17 39)(18 40)(19 33)(20 34)(21 35)(22 36)(23 37)(24 38)(25 45)(26 46)(27 47)(28 48)(29 41)(30 42)(31 43)(32 44)
(1 35 30)(2 31 36)(3 37 32)(4 25 38)(5 39 26)(6 27 40)(7 33 28)(8 29 34)(9 41 20)(10 21 42)(11 43 22)(12 23 44)(13 45 24)(14 17 46)(15 47 18)(16 19 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)
G:=sub<Sym(48)| (1,30,35)(2,31,36)(3,32,37)(4,25,38)(5,26,39)(6,27,40)(7,28,33)(8,29,34)(9,20,41)(10,21,42)(11,22,43)(12,23,44)(13,24,45)(14,17,46)(15,18,47)(16,19,48), (1,10)(2,11)(3,12)(4,13)(5,14)(6,15)(7,16)(8,9)(17,39)(18,40)(19,33)(20,34)(21,35)(22,36)(23,37)(24,38)(25,45)(26,46)(27,47)(28,48)(29,41)(30,42)(31,43)(32,44), (1,35,30)(2,31,36)(3,37,32)(4,25,38)(5,39,26)(6,27,40)(7,33,28)(8,29,34)(9,41,20)(10,21,42)(11,43,22)(12,23,44)(13,45,24)(14,17,46)(15,47,18)(16,19,48), (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)>;
G:=Group( (1,30,35)(2,31,36)(3,32,37)(4,25,38)(5,26,39)(6,27,40)(7,28,33)(8,29,34)(9,20,41)(10,21,42)(11,22,43)(12,23,44)(13,24,45)(14,17,46)(15,18,47)(16,19,48), (1,10)(2,11)(3,12)(4,13)(5,14)(6,15)(7,16)(8,9)(17,39)(18,40)(19,33)(20,34)(21,35)(22,36)(23,37)(24,38)(25,45)(26,46)(27,47)(28,48)(29,41)(30,42)(31,43)(32,44), (1,35,30)(2,31,36)(3,37,32)(4,25,38)(5,39,26)(6,27,40)(7,33,28)(8,29,34)(9,41,20)(10,21,42)(11,43,22)(12,23,44)(13,45,24)(14,17,46)(15,47,18)(16,19,48), (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) );
G=PermutationGroup([(1,30,35),(2,31,36),(3,32,37),(4,25,38),(5,26,39),(6,27,40),(7,28,33),(8,29,34),(9,20,41),(10,21,42),(11,22,43),(12,23,44),(13,24,45),(14,17,46),(15,18,47),(16,19,48)], [(1,10),(2,11),(3,12),(4,13),(5,14),(6,15),(7,16),(8,9),(17,39),(18,40),(19,33),(20,34),(21,35),(22,36),(23,37),(24,38),(25,45),(26,46),(27,47),(28,48),(29,41),(30,42),(31,43),(32,44)], [(1,35,30),(2,31,36),(3,37,32),(4,25,38),(5,39,26),(6,27,40),(7,33,28),(8,29,34),(9,41,20),(10,21,42),(11,43,22),(12,23,44),(13,45,24),(14,17,46),(15,47,18),(16,19,48)], [(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)])
S3×C3⋊C8 is a maximal subgroup of
S32×C8 C24.64D6 C24.D6 D12.2Dic3 D12.Dic3 D12.22D6 Dic6.20D6 D12.12D6 D12.13D6 C32⋊C6⋊C8 C12.89S32 C12.93S32
S3×C3⋊C8 is a maximal quotient of
C24.61D6 C12.77D12 C12.81D12 C32⋊C6⋊C8 C12.93S32
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 6D | 6E | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 24A | 24B | 24C | 24D |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 3 | 3 | 2 | 2 | 4 | 1 | 1 | 3 | 3 | 2 | 2 | 4 | 6 | 6 | 3 | 3 | 3 | 3 | 9 | 9 | 9 | 9 | 2 | 2 | 2 | 2 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | - | + | - | |||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C8 | S3 | S3 | Dic3 | D6 | Dic3 | C3⋊C8 | C4×S3 | S3×C8 | S32 | S3×Dic3 | S3×C3⋊C8 |
| kernel | S3×C3⋊C8 | C3×C3⋊C8 | C32⋊4C8 | S3×C12 | C3×Dic3 | S3×C6 | C3×S3 | C3⋊C8 | C4×S3 | Dic3 | C12 | D6 | S3 | C6 | C3 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 1 | 1 | 1 | 2 | 1 | 4 | 2 | 4 | 1 | 1 | 2 |
Matrix representation of S3×C3⋊C8 ►in GL4(𝔽5) generated by
| 1 | 2 | 0 | 0 |
| 1 | 3 | 0 | 0 |
| 0 | 0 | 3 | 3 |
| 0 | 0 | 4 | 1 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 1 | 2 | 0 | 0 |
| 1 | 3 | 0 | 0 |
| 0 | 0 | 1 | 2 |
| 0 | 0 | 1 | 3 |
| 0 | 0 | 0 | 3 |
| 0 | 0 | 1 | 0 |
| 0 | 3 | 0 | 0 |
| 1 | 0 | 0 | 0 |
G:=sub<GL(4,GF(5))| [1,1,0,0,2,3,0,0,0,0,3,4,0,0,3,1],[0,0,1,0,0,0,0,1,1,0,0,0,0,1,0,0],[1,1,0,0,2,3,0,0,0,0,1,1,0,0,2,3],[0,0,0,1,0,0,3,0,0,1,0,0,3,0,0,0] >;
S3×C3⋊C8 in GAP, Magma, Sage, TeX
S_3\times C_3\rtimes C_8
% in TeX
G:=Group("S3xC3:C8"); // GroupNames label
G:=SmallGroup(144,52);
// by ID
G=gap.SmallGroup(144,52);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-3,31,50,490,3461]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^2=c^3=d^8=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations
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