direct product, metabelian, supersoluble, monomial, A-group
Aliases: C32×D9, C33.5S3, C9⋊3(C3×C6), (C3×C9)⋊14C6, (C32×C9)⋊3C2, C3.1(S3×C32), C32.15(C3×S3), SmallGroup(162,32)
Series: Derived ►Chief ►Lower central ►Upper central
| C9 — C32×D9 |
Generators and relations for C32×D9
G = < a,b,c,d | a3=b3=c9=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 128 in 52 conjugacy classes, 24 normal (8 characteristic)
C1, C2, C3, C3, C3, S3, C6, C9, C9, C32, C32, C32, D9, C3×S3, C3×C6, C3×C9, C3×C9, C33, C3×D9, S3×C32, C32×C9, C32×D9
Quotients: C1, C2, C3, S3, C6, C32, D9, C3×S3, C3×C6, C3×D9, S3×C32, C32×D9
►On 54 points
Generators in S54
(1 17 20)(2 18 21)(3 10 22)(4 11 23)(5 12 24)(6 13 25)(7 14 26)(8 15 27)(9 16 19)(28 43 49)(29 44 50)(30 45 51)(31 37 52)(32 38 53)(33 39 54)(34 40 46)(35 41 47)(36 42 48)
(1 23 14)(2 24 15)(3 25 16)(4 26 17)(5 27 18)(6 19 10)(7 20 11)(8 21 12)(9 22 13)(28 46 37)(29 47 38)(30 48 39)(31 49 40)(32 50 41)(33 51 42)(34 52 43)(35 53 44)(36 54 45)
(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 30)(2 29)(3 28)(4 36)(5 35)(6 34)(7 33)(8 32)(9 31)(10 43)(11 42)(12 41)(13 40)(14 39)(15 38)(16 37)(17 45)(18 44)(19 52)(20 51)(21 50)(22 49)(23 48)(24 47)(25 46)(26 54)(27 53)
G:=sub<Sym(54)| (1,17,20)(2,18,21)(3,10,22)(4,11,23)(5,12,24)(6,13,25)(7,14,26)(8,15,27)(9,16,19)(28,43,49)(29,44,50)(30,45,51)(31,37,52)(32,38,53)(33,39,54)(34,40,46)(35,41,47)(36,42,48), (1,23,14)(2,24,15)(3,25,16)(4,26,17)(5,27,18)(6,19,10)(7,20,11)(8,21,12)(9,22,13)(28,46,37)(29,47,38)(30,48,39)(31,49,40)(32,50,41)(33,51,42)(34,52,43)(35,53,44)(36,54,45), (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), (1,30)(2,29)(3,28)(4,36)(5,35)(6,34)(7,33)(8,32)(9,31)(10,43)(11,42)(12,41)(13,40)(14,39)(15,38)(16,37)(17,45)(18,44)(19,52)(20,51)(21,50)(22,49)(23,48)(24,47)(25,46)(26,54)(27,53)>;
G:=Group( (1,17,20)(2,18,21)(3,10,22)(4,11,23)(5,12,24)(6,13,25)(7,14,26)(8,15,27)(9,16,19)(28,43,49)(29,44,50)(30,45,51)(31,37,52)(32,38,53)(33,39,54)(34,40,46)(35,41,47)(36,42,48), (1,23,14)(2,24,15)(3,25,16)(4,26,17)(5,27,18)(6,19,10)(7,20,11)(8,21,12)(9,22,13)(28,46,37)(29,47,38)(30,48,39)(31,49,40)(32,50,41)(33,51,42)(34,52,43)(35,53,44)(36,54,45), (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), (1,30)(2,29)(3,28)(4,36)(5,35)(6,34)(7,33)(8,32)(9,31)(10,43)(11,42)(12,41)(13,40)(14,39)(15,38)(16,37)(17,45)(18,44)(19,52)(20,51)(21,50)(22,49)(23,48)(24,47)(25,46)(26,54)(27,53) );
G=PermutationGroup([[(1,17,20),(2,18,21),(3,10,22),(4,11,23),(5,12,24),(6,13,25),(7,14,26),(8,15,27),(9,16,19),(28,43,49),(29,44,50),(30,45,51),(31,37,52),(32,38,53),(33,39,54),(34,40,46),(35,41,47),(36,42,48)], [(1,23,14),(2,24,15),(3,25,16),(4,26,17),(5,27,18),(6,19,10),(7,20,11),(8,21,12),(9,22,13),(28,46,37),(29,47,38),(30,48,39),(31,49,40),(32,50,41),(33,51,42),(34,52,43),(35,53,44),(36,54,45)], [(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)], [(1,30),(2,29),(3,28),(4,36),(5,35),(6,34),(7,33),(8,32),(9,31),(10,43),(11,42),(12,41),(13,40),(14,39),(15,38),(16,37),(17,45),(18,44),(19,52),(20,51),(21,50),(22,49),(23,48),(24,47),(25,46),(26,54),(27,53)]])
C32×D9 is a maximal subgroup of
D9⋊He3 D9⋊3- 1+2 (C32×C9)⋊S3
54 conjugacy classes
| class | 1 | 2 | 3A | ··· | 3H | 3I | ··· | 3Q | 6A | ··· | 6H | 9A | ··· | 9AA |
| order | 1 | 2 | 3 | ··· | 3 | 3 | ··· | 3 | 6 | ··· | 6 | 9 | ··· | 9 |
| size | 1 | 9 | 1 | ··· | 1 | 2 | ··· | 2 | 9 | ··· | 9 | 2 | ··· | 2 |
54 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | ||||
| image | C1 | C2 | C3 | C6 | S3 | D9 | C3×S3 | C3×D9 |
| kernel | C32×D9 | C32×C9 | C3×D9 | C3×C9 | C33 | C32 | C32 | C3 |
| # reps | 1 | 1 | 8 | 8 | 1 | 3 | 8 | 24 |
Matrix representation of C32×D9 ►in GL3(𝔽19) generated by
| 7 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 0 | 7 | 0 |
| 0 | 0 | 7 |
| 1 | 0 | 0 |
| 0 | 9 | 3 |
| 0 | 0 | 17 |
| 18 | 0 | 0 |
| 0 | 6 | 18 |
| 0 | 16 | 13 |
G:=sub<GL(3,GF(19))| [7,0,0,0,1,0,0,0,1],[1,0,0,0,7,0,0,0,7],[1,0,0,0,9,0,0,3,17],[18,0,0,0,6,16,0,18,13] >;
C32×D9 in GAP, Magma, Sage, TeX
C_3^2\times D_9
% in TeX
G:=Group("C3^2xD9"); // GroupNames label
G:=SmallGroup(162,32);
// by ID
G=gap.SmallGroup(162,32);
# by ID
G:=PCGroup([5,-2,-3,-3,-3,-3,1803,138,2704]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^3=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