non-abelian, supersoluble, monomial
Aliases: (C3×C9)⋊2D9, C32⋊C9.5C6, (C32×C9).2S3, C32.6(C3×D9), C32.3(C9⋊C6), C33.27(C3×S3), C32⋊2D9.4C3, C3.7(C32⋊D9), C32.19He3⋊2C2, C32.37(C32⋊C6), C3.4(He3.C6), SmallGroup(486,21)
Series: Derived ►Chief ►Lower central ►Upper central
| C32⋊C9 — (C3×C9)⋊D9 |
Generators and relations for (C3×C9)⋊D9
G = < a,b,c,d | a3=b9=c9=d2=1, ab=ba, cac-1=ab6, dad=a-1b6, cbc-1=a-1b, bd=db, dcd=c-1 >
Smallest permutation representation of (C3×C9)⋊D9
►On 54 points
Generators in S54
(10 13 16)(11 14 17)(12 15 18)(19 25 22)(20 26 23)(21 27 24)(28 31 34)(29 32 35)(30 33 36)(37 43 40)(38 44 41)(39 45 42)
(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 42 35 4 45 29 7 39 32)(2 43 33 5 37 36 8 40 30)(3 44 31 6 38 34 9 41 28)(10 54 25 16 51 22 13 48 19)(11 49 20 17 46 26 14 52 23)(12 53 24 18 50 21 15 47 27)
(1 20)(2 21)(3 22)(4 23)(5 24)(6 25)(7 26)(8 27)(9 19)(10 34)(11 35)(12 36)(13 28)(14 29)(15 30)(16 31)(17 32)(18 33)(37 53)(38 54)(39 46)(40 47)(41 48)(42 49)(43 50)(44 51)(45 52)
G:=sub<Sym(54)| (10,13,16)(11,14,17)(12,15,18)(19,25,22)(20,26,23)(21,27,24)(28,31,34)(29,32,35)(30,33,36)(37,43,40)(38,44,41)(39,45,42), (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), (1,42,35,4,45,29,7,39,32)(2,43,33,5,37,36,8,40,30)(3,44,31,6,38,34,9,41,28)(10,54,25,16,51,22,13,48,19)(11,49,20,17,46,26,14,52,23)(12,53,24,18,50,21,15,47,27), (1,20)(2,21)(3,22)(4,23)(5,24)(6,25)(7,26)(8,27)(9,19)(10,34)(11,35)(12,36)(13,28)(14,29)(15,30)(16,31)(17,32)(18,33)(37,53)(38,54)(39,46)(40,47)(41,48)(42,49)(43,50)(44,51)(45,52)>;
G:=Group( (10,13,16)(11,14,17)(12,15,18)(19,25,22)(20,26,23)(21,27,24)(28,31,34)(29,32,35)(30,33,36)(37,43,40)(38,44,41)(39,45,42), (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), (1,42,35,4,45,29,7,39,32)(2,43,33,5,37,36,8,40,30)(3,44,31,6,38,34,9,41,28)(10,54,25,16,51,22,13,48,19)(11,49,20,17,46,26,14,52,23)(12,53,24,18,50,21,15,47,27), (1,20)(2,21)(3,22)(4,23)(5,24)(6,25)(7,26)(8,27)(9,19)(10,34)(11,35)(12,36)(13,28)(14,29)(15,30)(16,31)(17,32)(18,33)(37,53)(38,54)(39,46)(40,47)(41,48)(42,49)(43,50)(44,51)(45,52) );
G=PermutationGroup([[(10,13,16),(11,14,17),(12,15,18),(19,25,22),(20,26,23),(21,27,24),(28,31,34),(29,32,35),(30,33,36),(37,43,40),(38,44,41),(39,45,42)], [(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)], [(1,42,35,4,45,29,7,39,32),(2,43,33,5,37,36,8,40,30),(3,44,31,6,38,34,9,41,28),(10,54,25,16,51,22,13,48,19),(11,49,20,17,46,26,14,52,23),(12,53,24,18,50,21,15,47,27)], [(1,20),(2,21),(3,22),(4,23),(5,24),(6,25),(7,26),(8,27),(9,19),(10,34),(11,35),(12,36),(13,28),(14,29),(15,30),(16,31),(17,32),(18,33),(37,53),(38,54),(39,46),(40,47),(41,48),(42,49),(43,50),(44,51),(45,52)]])
39 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 3F | 3G | 3H | 6A | 6B | 9A | ··· | 9F | 9G | ··· | 9L | 9M | ··· | 9U | 18A | ··· | 18F |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 6 | 6 | 9 | ··· | 9 | 9 | ··· | 9 | 9 | ··· | 9 | 18 | ··· | 18 |
| size | 1 | 27 | 1 | 1 | 2 | 2 | 2 | 6 | 6 | 6 | 27 | 27 | 3 | ··· | 3 | 6 | ··· | 6 | 18 | ··· | 18 | 27 | ··· | 27 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 3 | 6 | 6 | 6 |
| type | + | + | + | + | + | + | ||||||
| image | C1 | C2 | C3 | C6 | S3 | D9 | C3×S3 | C3×D9 | He3.C6 | C32⋊C6 | C9⋊C6 | (C3×C9)⋊D9 |
| kernel | (C3×C9)⋊D9 | C32.19He3 | C32⋊2D9 | C32⋊C9 | C32×C9 | C3×C9 | C33 | C32 | C3 | C32 | C32 | C1 |
| # reps | 1 | 1 | 2 | 2 | 1 | 3 | 2 | 6 | 12 | 1 | 2 | 6 |
Matrix representation of (C3×C9)⋊D9 ►in GL5(𝔽19)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 11 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 7 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 17 | 0 | 0 |
| 0 | 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 0 | 5 |
| 17 | 4 | 0 | 0 | 0 |
| 18 | 11 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 |
| 0 | 0 | 5 | 0 | 0 |
| 0 | 0 | 0 | 5 | 0 |
| 0 | 2 | 0 | 0 | 0 |
| 10 | 0 | 0 | 0 | 0 |
| 0 | 0 | 18 | 0 | 0 |
| 0 | 0 | 0 | 0 | 15 |
| 0 | 0 | 0 | 14 | 0 |
G:=sub<GL(5,GF(19))| [1,0,0,0,0,0,1,0,0,0,0,0,11,0,0,0,0,0,1,0,0,0,0,0,7],[1,0,0,0,0,0,1,0,0,0,0,0,17,0,0,0,0,0,5,0,0,0,0,0,5],[17,18,0,0,0,4,11,0,0,0,0,0,0,5,0,0,0,0,0,5,0,0,16,0,0],[0,10,0,0,0,2,0,0,0,0,0,0,18,0,0,0,0,0,0,14,0,0,0,15,0] >;
(C3×C9)⋊D9 in GAP, Magma, Sage, TeX
(C_3\times C_9)\rtimes D_9
% in TeX
G:=Group("(C3xC9):D9"); // GroupNames label
G:=SmallGroup(486,21);
// by ID
G=gap.SmallGroup(486,21);
# by ID
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,979,1190,224,338,8643,1383,3244]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^9=c^9=d^2=1,a*b=b*a,c*a*c^-1=a*b^6,d*a*d=a^-1*b^6,c*b*c^-1=a^-1*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations
Export