direct product, metabelian, supersoluble, monomial, A-group
Aliases: D9×C3×C9, C92⋊23C6, C9⋊5(C3×C18), (C3×C92)⋊2C2, (C3×C9)⋊14C18, C3.4(C32×D9), (C32×C9).22S3, C32.15(S3×C9), C33.74(C3×S3), (C32×C9).29C6, (C32×D9).4C3, (C3×D9).7C32, C32.20(C3×D9), C32.27(S3×C32), C3.1(S3×C3×C9), (C3×C9).39(C3×C6), (C3×C9).50(C3×S3), SmallGroup(486,91)
Series: Derived ►Chief ►Lower central ►Upper central
| C9 — D9×C3×C9 |
Generators and relations for D9×C3×C9
G = < a,b,c,d | a3=b9=c9=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 256 in 108 conjugacy classes, 40 normal (16 characteristic)
C1, C2, C3, C3, C3, S3, C6, C9, C9, C9, C32, C32, C32, D9, C18, C3×S3, C3×C6, C3×C9, C3×C9, C3×C9, C33, C3×D9, C3×D9, S3×C9, C3×C18, S3×C32, C92, C92, C32×C9, C32×C9, C9×D9, C32×D9, S3×C3×C9, C3×C92, D9×C3×C9
Quotients: C1, C2, C3, S3, C6, C9, C32, D9, C18, C3×S3, C3×C6, C3×C9, C3×D9, S3×C9, C3×C18, S3×C32, C9×D9, C32×D9, S3×C3×C9, D9×C3×C9
►On 54 points
(1 29 45)(2 30 37)(3 31 38)(4 32 39)(5 33 40)(6 34 41)(7 35 42)(8 36 43)(9 28 44)(10 53 20)(11 54 21)(12 46 22)(13 47 23)(14 48 24)(15 49 25)(16 50 26)(17 51 27)(18 52 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 49 50 51 52 53 54)
(1 41 30 7 38 36 4 44 33)(2 42 31 8 39 28 5 45 34)(3 43 32 9 40 29 6 37 35)(10 48 19 13 51 22 16 54 25)(11 49 20 14 52 23 17 46 26)(12 50 21 15 53 24 18 47 27)
(1 26)(2 27)(3 19)(4 20)(5 21)(6 22)(7 23)(8 24)(9 25)(10 32)(11 33)(12 34)(13 35)(14 36)(15 28)(16 29)(17 30)(18 31)(37 51)(38 52)(39 53)(40 54)(41 46)(42 47)(43 48)(44 49)(45 50)
G:=sub<Sym(54)| (1,29,45)(2,30,37)(3,31,38)(4,32,39)(5,33,40)(6,34,41)(7,35,42)(8,36,43)(9,28,44)(10,53,20)(11,54,21)(12,46,22)(13,47,23)(14,48,24)(15,49,25)(16,50,26)(17,51,27)(18,52,19), (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)(46,47,48,49,50,51,52,53,54), (1,41,30,7,38,36,4,44,33)(2,42,31,8,39,28,5,45,34)(3,43,32,9,40,29,6,37,35)(10,48,19,13,51,22,16,54,25)(11,49,20,14,52,23,17,46,26)(12,50,21,15,53,24,18,47,27), (1,26)(2,27)(3,19)(4,20)(5,21)(6,22)(7,23)(8,24)(9,25)(10,32)(11,33)(12,34)(13,35)(14,36)(15,28)(16,29)(17,30)(18,31)(37,51)(38,52)(39,53)(40,54)(41,46)(42,47)(43,48)(44,49)(45,50)>;
G:=Group( (1,29,45)(2,30,37)(3,31,38)(4,32,39)(5,33,40)(6,34,41)(7,35,42)(8,36,43)(9,28,44)(10,53,20)(11,54,21)(12,46,22)(13,47,23)(14,48,24)(15,49,25)(16,50,26)(17,51,27)(18,52,19), (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)(46,47,48,49,50,51,52,53,54), (1,41,30,7,38,36,4,44,33)(2,42,31,8,39,28,5,45,34)(3,43,32,9,40,29,6,37,35)(10,48,19,13,51,22,16,54,25)(11,49,20,14,52,23,17,46,26)(12,50,21,15,53,24,18,47,27), (1,26)(2,27)(3,19)(4,20)(5,21)(6,22)(7,23)(8,24)(9,25)(10,32)(11,33)(12,34)(13,35)(14,36)(15,28)(16,29)(17,30)(18,31)(37,51)(38,52)(39,53)(40,54)(41,46)(42,47)(43,48)(44,49)(45,50) );
G=PermutationGroup([[(1,29,45),(2,30,37),(3,31,38),(4,32,39),(5,33,40),(6,34,41),(7,35,42),(8,36,43),(9,28,44),(10,53,20),(11,54,21),(12,46,22),(13,47,23),(14,48,24),(15,49,25),(16,50,26),(17,51,27),(18,52,19)], [(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),(46,47,48,49,50,51,52,53,54)], [(1,41,30,7,38,36,4,44,33),(2,42,31,8,39,28,5,45,34),(3,43,32,9,40,29,6,37,35),(10,48,19,13,51,22,16,54,25),(11,49,20,14,52,23,17,46,26),(12,50,21,15,53,24,18,47,27)], [(1,26),(2,27),(3,19),(4,20),(5,21),(6,22),(7,23),(8,24),(9,25),(10,32),(11,33),(12,34),(13,35),(14,36),(15,28),(16,29),(17,30),(18,31),(37,51),(38,52),(39,53),(40,54),(41,46),(42,47),(43,48),(44,49),(45,50)]])
162 conjugacy classes
| class | 1 | 2 | 3A | ··· | 3H | 3I | ··· | 3Q | 6A | ··· | 6H | 9A | ··· | 9R | 9S | ··· | 9DM | 18A | ··· | 18R |
| order | 1 | 2 | 3 | ··· | 3 | 3 | ··· | 3 | 6 | ··· | 6 | 9 | ··· | 9 | 9 | ··· | 9 | 18 | ··· | 18 |
| size | 1 | 9 | 1 | ··· | 1 | 2 | ··· | 2 | 9 | ··· | 9 | 1 | ··· | 1 | 2 | ··· | 2 | 9 | ··· | 9 |
162 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | ||||||||||||
| image | C1 | C2 | C3 | C3 | C6 | C6 | C9 | C18 | S3 | D9 | C3×S3 | C3×S3 | C3×D9 | C3×D9 | S3×C9 | C9×D9 |
| kernel | D9×C3×C9 | C3×C92 | C9×D9 | C32×D9 | C92 | C32×C9 | C3×D9 | C3×C9 | C32×C9 | C3×C9 | C3×C9 | C33 | C9 | C32 | C32 | C3 |
| # reps | 1 | 1 | 6 | 2 | 6 | 2 | 18 | 18 | 1 | 3 | 6 | 2 | 18 | 6 | 18 | 54 |
Matrix representation of D9×C3×C9 ►in GL3(𝔽19) generated by
| 11 | 0 | 0 |
| 0 | 7 | 0 |
| 0 | 0 | 7 |
| 11 | 0 | 0 |
| 0 | 9 | 0 |
| 0 | 0 | 9 |
| 1 | 0 | 0 |
| 0 | 5 | 0 |
| 0 | 0 | 4 |
| 1 | 0 | 0 |
| 0 | 0 | 4 |
| 0 | 5 | 0 |
G:=sub<GL(3,GF(19))| [11,0,0,0,7,0,0,0,7],[11,0,0,0,9,0,0,0,9],[1,0,0,0,5,0,0,0,4],[1,0,0,0,0,5,0,4,0] >;
D9×C3×C9 in GAP, Magma, Sage, TeX
D_9\times C_3\times C_9
% in TeX
G:=Group("D9xC3xC9"); // GroupNames label
G:=SmallGroup(486,91);
// by ID
G=gap.SmallGroup(486,91);
# by ID
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,68,8104,208,11669]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^9=c^9=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