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 25 35)(2 26 36)(3 27 37)(4 28 38)(5 29 39)(6 30 40)(7 31 33)(8 32 34)(9 23 44)(10 24 45)(11 17 46)(12 18 47)(13 19 48)(14 20 41)(15 21 42)(16 22 43)
(1 15)(2 16)(3 9)(4 10)(5 11)(6 12)(7 13)(8 14)(17 39)(18 40)(19 33)(20 34)(21 35)(22 36)(23 37)(24 38)(25 42)(26 43)(27 44)(28 45)(29 46)(30 47)(31 48)(32 41)
(1 35 25)(2 26 36)(3 37 27)(4 28 38)(5 39 29)(6 30 40)(7 33 31)(8 32 34)(9 23 44)(10 45 24)(11 17 46)(12 47 18)(13 19 48)(14 41 20)(15 21 42)(16 43 22)
(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)| (1,25,35)(2,26,36)(3,27,37)(4,28,38)(5,29,39)(6,30,40)(7,31,33)(8,32,34)(9,23,44)(10,24,45)(11,17,46)(12,18,47)(13,19,48)(14,20,41)(15,21,42)(16,22,43), (1,15)(2,16)(3,9)(4,10)(5,11)(6,12)(7,13)(8,14)(17,39)(18,40)(19,33)(20,34)(21,35)(22,36)(23,37)(24,38)(25,42)(26,43)(27,44)(28,45)(29,46)(30,47)(31,48)(32,41), (1,35,25)(2,26,36)(3,37,27)(4,28,38)(5,39,29)(6,30,40)(7,33,31)(8,32,34)(9,23,44)(10,45,24)(11,17,46)(12,47,18)(13,19,48)(14,41,20)(15,21,42)(16,43,22), (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( (1,25,35)(2,26,36)(3,27,37)(4,28,38)(5,29,39)(6,30,40)(7,31,33)(8,32,34)(9,23,44)(10,24,45)(11,17,46)(12,18,47)(13,19,48)(14,20,41)(15,21,42)(16,22,43), (1,15)(2,16)(3,9)(4,10)(5,11)(6,12)(7,13)(8,14)(17,39)(18,40)(19,33)(20,34)(21,35)(22,36)(23,37)(24,38)(25,42)(26,43)(27,44)(28,45)(29,46)(30,47)(31,48)(32,41), (1,35,25)(2,26,36)(3,37,27)(4,28,38)(5,39,29)(6,30,40)(7,33,31)(8,32,34)(9,23,44)(10,45,24)(11,17,46)(12,47,18)(13,19,48)(14,41,20)(15,21,42)(16,43,22), (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([[(1,25,35),(2,26,36),(3,27,37),(4,28,38),(5,29,39),(6,30,40),(7,31,33),(8,32,34),(9,23,44),(10,24,45),(11,17,46),(12,18,47),(13,19,48),(14,20,41),(15,21,42),(16,22,43)], [(1,15),(2,16),(3,9),(4,10),(5,11),(6,12),(7,13),(8,14),(17,39),(18,40),(19,33),(20,34),(21,35),(22,36),(23,37),(24,38),(25,42),(26,43),(27,44),(28,45),(29,46),(30,47),(31,48),(32,41)], [(1,35,25),(2,26,36),(3,37,27),(4,28,38),(5,39,29),(6,30,40),(7,33,31),(8,32,34),(9,23,44),(10,45,24),(11,17,46),(12,47,18),(13,19,48),(14,41,20),(15,21,42),(16,43,22)], [(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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