direct product, metabelian, supersoluble, monomial, A-group
Aliases: S3×C3×C9, C32⋊3C18, C33.5C6, C3⋊(C3×C18), (C3×C9)⋊13C6, (C32×C9)⋊1C2, C3.4(S3×C32), C32.6(C3×C6), (C3×S3).1C32, (S3×C32).2C3, C32.19(C3×S3), SmallGroup(162,33)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — S3×C3×C9 |
Generators and relations for S3×C3×C9
G = < a,b,c,d | a3=b9=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 90 in 55 conjugacy classes, 30 normal (12 characteristic)
C1, C2, C3, C3, C3, S3, C6, C9, C9, C32, C32, C32, C18, C3×S3, C3×S3, C3×C6, C3×C9, C3×C9, C3×C9, C33, S3×C9, C3×C18, S3×C32, C32×C9, S3×C3×C9
Quotients: C1, C2, C3, S3, C6, C9, C32, C18, C3×S3, C3×C6, C3×C9, S3×C9, C3×C18, S3×C32, S3×C3×C9
►On 54 points
Generators in S54
(1 31 41)(2 32 42)(3 33 43)(4 34 44)(5 35 45)(6 36 37)(7 28 38)(8 29 39)(9 30 40)(10 20 51)(11 21 52)(12 22 53)(13 23 54)(14 24 46)(15 25 47)(16 26 48)(17 27 49)(18 19 50)
(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 49 50 51 52 53 54)
(1 28 44)(2 29 45)(3 30 37)(4 31 38)(5 32 39)(6 33 40)(7 34 41)(8 35 42)(9 36 43)(10 54 26)(11 46 27)(12 47 19)(13 48 20)(14 49 21)(15 50 22)(16 51 23)(17 52 24)(18 53 25)
(1 52)(2 53)(3 54)(4 46)(5 47)(6 48)(7 49)(8 50)(9 51)(10 30)(11 31)(12 32)(13 33)(14 34)(15 35)(16 36)(17 28)(18 29)(19 39)(20 40)(21 41)(22 42)(23 43)(24 44)(25 45)(26 37)(27 38)
G:=sub<Sym(54)| (1,31,41)(2,32,42)(3,33,43)(4,34,44)(5,35,45)(6,36,37)(7,28,38)(8,29,39)(9,30,40)(10,20,51)(11,21,52)(12,22,53)(13,23,54)(14,24,46)(15,25,47)(16,26,48)(17,27,49)(18,19,50), (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)(46,47,48,49,50,51,52,53,54), (1,28,44)(2,29,45)(3,30,37)(4,31,38)(5,32,39)(6,33,40)(7,34,41)(8,35,42)(9,36,43)(10,54,26)(11,46,27)(12,47,19)(13,48,20)(14,49,21)(15,50,22)(16,51,23)(17,52,24)(18,53,25), (1,52)(2,53)(3,54)(4,46)(5,47)(6,48)(7,49)(8,50)(9,51)(10,30)(11,31)(12,32)(13,33)(14,34)(15,35)(16,36)(17,28)(18,29)(19,39)(20,40)(21,41)(22,42)(23,43)(24,44)(25,45)(26,37)(27,38)>;
G:=Group( (1,31,41)(2,32,42)(3,33,43)(4,34,44)(5,35,45)(6,36,37)(7,28,38)(8,29,39)(9,30,40)(10,20,51)(11,21,52)(12,22,53)(13,23,54)(14,24,46)(15,25,47)(16,26,48)(17,27,49)(18,19,50), (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)(46,47,48,49,50,51,52,53,54), (1,28,44)(2,29,45)(3,30,37)(4,31,38)(5,32,39)(6,33,40)(7,34,41)(8,35,42)(9,36,43)(10,54,26)(11,46,27)(12,47,19)(13,48,20)(14,49,21)(15,50,22)(16,51,23)(17,52,24)(18,53,25), (1,52)(2,53)(3,54)(4,46)(5,47)(6,48)(7,49)(8,50)(9,51)(10,30)(11,31)(12,32)(13,33)(14,34)(15,35)(16,36)(17,28)(18,29)(19,39)(20,40)(21,41)(22,42)(23,43)(24,44)(25,45)(26,37)(27,38) );
G=PermutationGroup([[(1,31,41),(2,32,42),(3,33,43),(4,34,44),(5,35,45),(6,36,37),(7,28,38),(8,29,39),(9,30,40),(10,20,51),(11,21,52),(12,22,53),(13,23,54),(14,24,46),(15,25,47),(16,26,48),(17,27,49),(18,19,50)], [(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),(46,47,48,49,50,51,52,53,54)], [(1,28,44),(2,29,45),(3,30,37),(4,31,38),(5,32,39),(6,33,40),(7,34,41),(8,35,42),(9,36,43),(10,54,26),(11,46,27),(12,47,19),(13,48,20),(14,49,21),(15,50,22),(16,51,23),(17,52,24),(18,53,25)], [(1,52),(2,53),(3,54),(4,46),(5,47),(6,48),(7,49),(8,50),(9,51),(10,30),(11,31),(12,32),(13,33),(14,34),(15,35),(16,36),(17,28),(18,29),(19,39),(20,40),(21,41),(22,42),(23,43),(24,44),(25,45),(26,37),(27,38)]])
S3×C3×C9 is a maximal subgroup of
He3⋊C18 C3≀S3⋊3C3
81 conjugacy classes
| class | 1 | 2 | 3A | ··· | 3H | 3I | ··· | 3Q | 6A | ··· | 6H | 9A | ··· | 9R | 9S | ··· | 9AJ | 18A | ··· | 18R |
| order | 1 | 2 | 3 | ··· | 3 | 3 | ··· | 3 | 6 | ··· | 6 | 9 | ··· | 9 | 9 | ··· | 9 | 18 | ··· | 18 |
| size | 1 | 3 | 1 | ··· | 1 | 2 | ··· | 2 | 3 | ··· | 3 | 1 | ··· | 1 | 2 | ··· | 2 | 3 | ··· | 3 |
81 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | |||||||||
| image | C1 | C2 | C3 | C3 | C6 | C6 | C9 | C18 | S3 | C3×S3 | C3×S3 | S3×C9 |
| kernel | S3×C3×C9 | C32×C9 | S3×C9 | S3×C32 | C3×C9 | C33 | C3×S3 | C32 | C3×C9 | C9 | C32 | C3 |
| # reps | 1 | 1 | 6 | 2 | 6 | 2 | 18 | 18 | 1 | 6 | 2 | 18 |
Matrix representation of S3×C3×C9 ►in GL3(𝔽19) generated by
| 11 | 0 | 0 |
| 0 | 7 | 0 |
| 0 | 0 | 7 |
| 6 | 0 | 0 |
| 0 | 11 | 0 |
| 0 | 0 | 11 |
| 1 | 0 | 0 |
| 0 | 7 | 0 |
| 0 | 8 | 11 |
| 1 | 0 | 0 |
| 0 | 1 | 10 |
| 0 | 0 | 18 |
G:=sub<GL(3,GF(19))| [11,0,0,0,7,0,0,0,7],[6,0,0,0,11,0,0,0,11],[1,0,0,0,7,8,0,0,11],[1,0,0,0,1,0,0,10,18] >;
S3×C3×C9 in GAP, Magma, Sage, TeX
S_3\times C_3\times C_9
% in TeX
G:=Group("S3xC3xC9"); // GroupNames label
G:=SmallGroup(162,33);
// by ID
G=gap.SmallGroup(162,33);
# by ID
G:=PCGroup([5,-2,-3,-3,-3,-3,57,2704]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^9=c^3=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations