non-abelian, supersoluble, monomial
Aliases: C32⋊C9.2S3, C32⋊C9.2C6, C33.7(C3×S3), C32⋊2D9.1C3, C32.29He3⋊1C2, C32.31(C32⋊C6), C3.4(He3.S3), C3.3(He3.2C6), SmallGroup(486,10)
Series: Derived ►Chief ►Lower central ►Upper central
| C32⋊C9 — C32⋊C9.C6 |
Generators and relations for C32⋊C9.C6
G = < a,b,c,d,e | a3=b3=c9=e2=1, d3=ebe=b-1, ab=ba, cac-1=ab-1, dad-1=ac3, eae=a-1c6, bc=cb, bd=db, dcd-1=a-1bc7, ce=ec, ede=bd2 >
Smallest permutation representation of C32⋊C9.C6
►On 54 points
Generators in S54
(1 7 4)(3 6 9)(11 17 14)(12 15 18)(19 25 22)(20 23 26)(28 34 31)(30 33 36)(38 41 44)(39 45 42)(46 52 49)(47 50 53)
(1 7 4)(2 8 5)(3 9 6)(10 13 16)(11 14 17)(12 15 18)(19 22 25)(20 23 26)(21 24 27)(28 34 31)(29 35 32)(30 36 33)(37 43 40)(38 44 41)(39 45 42)(46 49 52)(47 50 53)(48 51 54)
(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 32 44 4 35 38 7 29 41)(2 36 45 5 30 39 8 33 42)(3 34 43 6 28 37 9 31 40)(10 23 46 16 20 52 13 26 49)(11 24 50 17 21 47 14 27 53)(12 22 48 18 19 54 15 25 51)
(1 13)(2 14)(3 15)(4 16)(5 17)(6 18)(7 10)(8 11)(9 12)(19 43)(20 44)(21 45)(22 37)(23 38)(24 39)(25 40)(26 41)(27 42)(28 48)(29 49)(30 50)(31 51)(32 52)(33 53)(34 54)(35 46)(36 47)
G:=sub<Sym(54)| (1,7,4)(3,6,9)(11,17,14)(12,15,18)(19,25,22)(20,23,26)(28,34,31)(30,33,36)(38,41,44)(39,45,42)(46,52,49)(47,50,53), (1,7,4)(2,8,5)(3,9,6)(10,13,16)(11,14,17)(12,15,18)(19,22,25)(20,23,26)(21,24,27)(28,34,31)(29,35,32)(30,36,33)(37,43,40)(38,44,41)(39,45,42)(46,49,52)(47,50,53)(48,51,54), (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), (1,32,44,4,35,38,7,29,41)(2,36,45,5,30,39,8,33,42)(3,34,43,6,28,37,9,31,40)(10,23,46,16,20,52,13,26,49)(11,24,50,17,21,47,14,27,53)(12,22,48,18,19,54,15,25,51), (1,13)(2,14)(3,15)(4,16)(5,17)(6,18)(7,10)(8,11)(9,12)(19,43)(20,44)(21,45)(22,37)(23,38)(24,39)(25,40)(26,41)(27,42)(28,48)(29,49)(30,50)(31,51)(32,52)(33,53)(34,54)(35,46)(36,47)>;
G:=Group( (1,7,4)(3,6,9)(11,17,14)(12,15,18)(19,25,22)(20,23,26)(28,34,31)(30,33,36)(38,41,44)(39,45,42)(46,52,49)(47,50,53), (1,7,4)(2,8,5)(3,9,6)(10,13,16)(11,14,17)(12,15,18)(19,22,25)(20,23,26)(21,24,27)(28,34,31)(29,35,32)(30,36,33)(37,43,40)(38,44,41)(39,45,42)(46,49,52)(47,50,53)(48,51,54), (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), (1,32,44,4,35,38,7,29,41)(2,36,45,5,30,39,8,33,42)(3,34,43,6,28,37,9,31,40)(10,23,46,16,20,52,13,26,49)(11,24,50,17,21,47,14,27,53)(12,22,48,18,19,54,15,25,51), (1,13)(2,14)(3,15)(4,16)(5,17)(6,18)(7,10)(8,11)(9,12)(19,43)(20,44)(21,45)(22,37)(23,38)(24,39)(25,40)(26,41)(27,42)(28,48)(29,49)(30,50)(31,51)(32,52)(33,53)(34,54)(35,46)(36,47) );
G=PermutationGroup([[(1,7,4),(3,6,9),(11,17,14),(12,15,18),(19,25,22),(20,23,26),(28,34,31),(30,33,36),(38,41,44),(39,45,42),(46,52,49),(47,50,53)], [(1,7,4),(2,8,5),(3,9,6),(10,13,16),(11,14,17),(12,15,18),(19,22,25),(20,23,26),(21,24,27),(28,34,31),(29,35,32),(30,36,33),(37,43,40),(38,44,41),(39,45,42),(46,49,52),(47,50,53),(48,51,54)], [(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)], [(1,32,44,4,35,38,7,29,41),(2,36,45,5,30,39,8,33,42),(3,34,43,6,28,37,9,31,40),(10,23,46,16,20,52,13,26,49),(11,24,50,17,21,47,14,27,53),(12,22,48,18,19,54,15,25,51)], [(1,13),(2,14),(3,15),(4,16),(5,17),(6,18),(7,10),(8,11),(9,12),(19,43),(20,44),(21,45),(22,37),(23,38),(24,39),(25,40),(26,41),(27,42),(28,48),(29,49),(30,50),(31,51),(32,52),(33,53),(34,54),(35,46),(36,47)]])
31 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 3F | 6A | 6B | 9A | ··· | 9F | 9G | ··· | 9O | 18A | ··· | 18F |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | 6 | 6 | 9 | ··· | 9 | 9 | ··· | 9 | 18 | ··· | 18 |
| size | 1 | 27 | 1 | 1 | 2 | 2 | 2 | 18 | 27 | 27 | 9 | ··· | 9 | 18 | ··· | 18 | 27 | ··· | 27 |
31 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 3 | 6 | 6 | 6 |
| type | + | + | + | + | + | |||||
| image | C1 | C2 | C3 | C6 | S3 | C3×S3 | He3.2C6 | C32⋊C6 | He3.S3 | C32⋊C9.C6 |
| kernel | C32⋊C9.C6 | C32.29He3 | C32⋊2D9 | C32⋊C9 | C32⋊C9 | C33 | C3 | C32 | C3 | C1 |
| # reps | 1 | 1 | 2 | 2 | 1 | 2 | 12 | 1 | 3 | 6 |
Matrix representation of C32⋊C9.C6 ►in GL6(𝔽19)
| 11 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 11 | 8 | 7 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 8 | 0 | 18 | 11 | 0 |
| 12 | 8 | 0 | 11 | 0 | 7 |
| 11 | 0 | 0 | 0 | 0 | 0 |
| 0 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 0 | 0 | 0 |
| 0 | 0 | 0 | 7 | 0 | 0 |
| 11 | 12 | 0 | 0 | 7 | 0 |
| 12 | 1 | 0 | 0 | 0 | 7 |
| 10 | 13 | 5 | 0 | 0 | 0 |
| 6 | 0 | 0 | 0 | 0 | 0 |
| 10 | 4 | 9 | 0 | 0 | 0 |
| 4 | 13 | 0 | 6 | 0 | 5 |
| 6 | 0 | 9 | 13 | 0 | 10 |
| 1 | 4 | 15 | 9 | 9 | 13 |
| 8 | 18 | 4 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 8 | 18 | 11 | 0 | 0 | 0 |
| 12 | 1 | 0 | 7 | 6 | 0 |
| 8 | 18 | 11 | 0 | 12 | 1 |
| 7 | 11 | 12 | 8 | 8 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 12 | 1 | 0 | 7 | 6 | 0 |
| 0 | 0 | 0 | 0 | 11 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 18 | 0 |
| 0 | 0 | 1 | 0 | 11 | 0 |
G:=sub<GL(6,GF(19))| [11,0,11,0,0,12,0,1,8,0,8,8,0,0,7,0,0,0,0,0,0,1,18,11,0,0,0,0,11,0,0,0,0,0,0,7],[11,0,0,0,11,12,0,11,0,0,12,1,0,0,11,0,0,0,0,0,0,7,0,0,0,0,0,0,7,0,0,0,0,0,0,7],[10,6,10,4,6,1,13,0,4,13,0,4,5,0,9,0,9,15,0,0,0,6,13,9,0,0,0,0,0,9,0,0,0,5,10,13],[8,1,8,12,8,7,18,0,18,1,18,11,4,0,11,0,11,12,0,0,0,7,0,8,0,0,0,6,12,8,0,0,0,0,1,0],[0,12,0,1,0,0,0,1,0,0,0,0,0,0,0,0,0,1,1,7,0,0,0,0,0,6,11,0,18,11,0,0,1,0,0,0] >;
C32⋊C9.C6 in GAP, Magma, Sage, TeX
C_3^2\rtimes C_9.C_6
% in TeX
G:=Group("C3^2:C9.C6"); // GroupNames label
G:=SmallGroup(486,10);
// by ID
G=gap.SmallGroup(486,10);
# by ID
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,979,2162,224,176,4755,873,735,3244]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^3=c^9=e^2=1,d^3=e*b*e=b^-1,a*b=b*a,c*a*c^-1=a*b^-1,d*a*d^-1=a*c^3,e*a*e=a^-1*c^6,b*c=c*b,b*d=d*b,d*c*d^-1=a^-1*b*c^7,c*e=e*c,e*d*e=b*d^2>;
// generators/relations
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