non-abelian, supersoluble, monomial
Aliases: (C3×C9)⋊3D9, C32⋊C9.6C6, (C32×C9).3S3, C32.7(C3×D9), C32.4(C9⋊C6), C33.28(C3×S3), C32⋊2D9.5C3, C3.8(C32⋊D9), C32.20He3⋊2C2, C32.39(C32⋊C6), C3.4(He3.2C6), SmallGroup(486,23)
Series: Derived ►Chief ►Lower central ►Upper central
| C32⋊C9 — (C3×C9)⋊3D9 |
Generators and relations for (C3×C9)⋊3D9
G = < a,b,c,d | a3=b9=c9=d2=1, ab=ba, cac-1=ab3, dad=a-1b3, cbc-1=a-1b, bd=db, dcd=c-1 >
Smallest permutation representation of (C3×C9)⋊3D9
►On 54 points
Generators in S54
(19 22 25)(20 23 26)(21 24 27)(28 34 31)(29 35 32)(30 36 33)(37 40 43)(38 41 44)(39 42 45)(46 52 49)(47 53 50)(48 54 51)
(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 40 32 4 43 35 7 37 29)(2 41 36 5 44 30 8 38 33)(3 42 31 6 45 34 9 39 28)(10 20 52 16 26 49 13 23 46)(11 21 47 17 27 53 14 24 50)(12 22 51 18 19 48 15 25 54)
(1 26)(2 27)(3 19)(4 20)(5 21)(6 22)(7 23)(8 24)(9 25)(10 43)(11 44)(12 45)(13 37)(14 38)(15 39)(16 40)(17 41)(18 42)(28 48)(29 49)(30 50)(31 51)(32 52)(33 53)(34 54)(35 46)(36 47)
G:=sub<Sym(54)| (19,22,25)(20,23,26)(21,24,27)(28,34,31)(29,35,32)(30,36,33)(37,40,43)(38,41,44)(39,42,45)(46,52,49)(47,53,50)(48,54,51), (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), (1,40,32,4,43,35,7,37,29)(2,41,36,5,44,30,8,38,33)(3,42,31,6,45,34,9,39,28)(10,20,52,16,26,49,13,23,46)(11,21,47,17,27,53,14,24,50)(12,22,51,18,19,48,15,25,54), (1,26)(2,27)(3,19)(4,20)(5,21)(6,22)(7,23)(8,24)(9,25)(10,43)(11,44)(12,45)(13,37)(14,38)(15,39)(16,40)(17,41)(18,42)(28,48)(29,49)(30,50)(31,51)(32,52)(33,53)(34,54)(35,46)(36,47)>;
G:=Group( (19,22,25)(20,23,26)(21,24,27)(28,34,31)(29,35,32)(30,36,33)(37,40,43)(38,41,44)(39,42,45)(46,52,49)(47,53,50)(48,54,51), (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), (1,40,32,4,43,35,7,37,29)(2,41,36,5,44,30,8,38,33)(3,42,31,6,45,34,9,39,28)(10,20,52,16,26,49,13,23,46)(11,21,47,17,27,53,14,24,50)(12,22,51,18,19,48,15,25,54), (1,26)(2,27)(3,19)(4,20)(5,21)(6,22)(7,23)(8,24)(9,25)(10,43)(11,44)(12,45)(13,37)(14,38)(15,39)(16,40)(17,41)(18,42)(28,48)(29,49)(30,50)(31,51)(32,52)(33,53)(34,54)(35,46)(36,47) );
G=PermutationGroup([[(19,22,25),(20,23,26),(21,24,27),(28,34,31),(29,35,32),(30,36,33),(37,40,43),(38,41,44),(39,42,45),(46,52,49),(47,53,50),(48,54,51)], [(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)], [(1,40,32,4,43,35,7,37,29),(2,41,36,5,44,30,8,38,33),(3,42,31,6,45,34,9,39,28),(10,20,52,16,26,49,13,23,46),(11,21,47,17,27,53,14,24,50),(12,22,51,18,19,48,15,25,54)], [(1,26),(2,27),(3,19),(4,20),(5,21),(6,22),(7,23),(8,24),(9,25),(10,43),(11,44),(12,45),(13,37),(14,38),(15,39),(16,40),(17,41),(18,42),(28,48),(29,49),(30,50),(31,51),(32,52),(33,53),(34,54),(35,46),(36,47)]])
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 | He3.2C6 | C32⋊C6 | C9⋊C6 | (C3×C9)⋊3D9 |
| kernel | (C3×C9)⋊3D9 | C32.20He3 | 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)⋊3D9 ►in GL5(𝔽19)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 7 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 11 |
| 11 | 0 | 0 | 0 | 0 |
| 0 | 11 | 0 | 0 | 0 |
| 0 | 0 | 5 | 0 | 0 |
| 0 | 0 | 0 | 17 | 0 |
| 0 | 0 | 0 | 0 | 17 |
| 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 | 17 | 0 | 0 | 0 |
| 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 |
| 0 | 0 | 0 | 5 | 0 |
G:=sub<GL(5,GF(19))| [1,0,0,0,0,0,1,0,0,0,0,0,7,0,0,0,0,0,1,0,0,0,0,0,11],[11,0,0,0,0,0,11,0,0,0,0,0,5,0,0,0,0,0,17,0,0,0,0,0,17],[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],[0,9,0,0,0,17,0,0,0,0,0,0,1,0,0,0,0,0,0,5,0,0,0,4,0] >;
(C3×C9)⋊3D9 in GAP, Magma, Sage, TeX
(C_3\times C_9)\rtimes_3D_9
% in TeX
G:=Group("(C3xC9):3D9"); // GroupNames label
G:=SmallGroup(486,23);
// by ID
G=gap.SmallGroup(486,23);
# by ID
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,979,1190,224,338,4755,735,3244]);
// 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^3,d*a*d=a^-1*b^3,c*b*c^-1=a^-1*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations
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