non-abelian, supersoluble, monomial
Aliases: C92⋊4S3, (C3×C9)⋊8D9, C9.4(C9⋊S3), C92⋊3C3⋊2C2, C32.9(C3×D9), C32⋊C9.23C6, (C32×C9).21S3, C33.39(C3×S3), C32⋊2D9.9C3, C3.1(He3.4C6), C3.11(C3×C9⋊S3), (C3×C9).19(C3×S3), (C3×C9).16(C3⋊S3), C32.31(C3×C3⋊S3), SmallGroup(486,140)
Series: Derived ►Chief ►Lower central ►Upper central
| C32⋊C9 — C92⋊4S3 |
Generators and relations for C92⋊4S3
G = < a,b,c,d | a9=b9=c3=d2=1, ab=ba, ac=ca, ad=da, cbc-1=a6b, dbd=b-1, dcd=c-1 >
Subgroups: 326 in 78 conjugacy classes, 24 normal (12 characteristic)
C1, C2, C3, C3, S3, C6, C9, C9, C32, C32, C32, D9, C18, C3×S3, C3⋊S3, C3×C9, C3×C9, C3×C9, C33, C3×D9, S3×C9, C3×C3⋊S3, C92, C32⋊C9, C32⋊C9, C9⋊C9, C32×C9, C9×D9, C32⋊2D9, C9×C3⋊S3, C92⋊3C3, C92⋊4S3
Quotients: C1, C2, C3, S3, C6, D9, C3×S3, C3⋊S3, C3×D9, C9⋊S3, C3×C3⋊S3, C3×C9⋊S3, He3.4C6, C92⋊4S3
►On 54 points
(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 44 34 4 38 28 7 41 31)(2 45 35 5 39 29 8 42 32)(3 37 36 6 40 30 9 43 33)(10 52 23 16 49 20 13 46 26)(11 53 24 17 50 21 14 47 27)(12 54 25 18 51 22 15 48 19)
(19 25 22)(20 26 23)(21 27 24)(28 34 31)(29 35 32)(30 36 33)(37 40 43)(38 41 44)(39 42 45)(46 49 52)(47 50 53)(48 51 54)
(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 52)(29 53)(30 54)(31 46)(32 47)(33 48)(34 49)(35 50)(36 51)
G:=sub<Sym(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), (1,44,34,4,38,28,7,41,31)(2,45,35,5,39,29,8,42,32)(3,37,36,6,40,30,9,43,33)(10,52,23,16,49,20,13,46,26)(11,53,24,17,50,21,14,47,27)(12,54,25,18,51,22,15,48,19), (19,25,22)(20,26,23)(21,27,24)(28,34,31)(29,35,32)(30,36,33)(37,40,43)(38,41,44)(39,42,45)(46,49,52)(47,50,53)(48,51,54), (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)(28,52)(29,53)(30,54)(31,46)(32,47)(33,48)(34,49)(35,50)(36,51)>;
G:=Group( (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), (1,44,34,4,38,28,7,41,31)(2,45,35,5,39,29,8,42,32)(3,37,36,6,40,30,9,43,33)(10,52,23,16,49,20,13,46,26)(11,53,24,17,50,21,14,47,27)(12,54,25,18,51,22,15,48,19), (19,25,22)(20,26,23)(21,27,24)(28,34,31)(29,35,32)(30,36,33)(37,40,43)(38,41,44)(39,42,45)(46,49,52)(47,50,53)(48,51,54), (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)(28,52)(29,53)(30,54)(31,46)(32,47)(33,48)(34,49)(35,50)(36,51) );
G=PermutationGroup([[(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)], [(1,44,34,4,38,28,7,41,31),(2,45,35,5,39,29,8,42,32),(3,37,36,6,40,30,9,43,33),(10,52,23,16,49,20,13,46,26),(11,53,24,17,50,21,14,47,27),(12,54,25,18,51,22,15,48,19)], [(19,25,22),(20,26,23),(21,27,24),(28,34,31),(29,35,32),(30,36,33),(37,40,43),(38,41,44),(39,42,45),(46,49,52),(47,50,53),(48,51,54)], [(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),(28,52),(29,53),(30,54),(31,46),(32,47),(33,48),(34,49),(35,50),(36,51)]])
63 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 3F | 3G | 3H | 6A | 6B | 9A | ··· | 9F | 9G | ··· | 9L | 9M | ··· | 9AS | 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 | 1 | ··· | 1 | 2 | ··· | 2 | 6 | ··· | 6 | 27 | ··· | 27 |
63 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 6 |
| type | + | + | + | + | + | |||||||
| image | C1 | C2 | C3 | C6 | S3 | S3 | D9 | C3×S3 | C3×S3 | C3×D9 | He3.4C6 | C92⋊4S3 |
| kernel | C92⋊4S3 | C92⋊3C3 | C32⋊2D9 | C32⋊C9 | C92 | C32×C9 | C3×C9 | C3×C9 | C33 | C32 | C3 | C1 |
| # reps | 1 | 1 | 2 | 2 | 3 | 1 | 9 | 6 | 2 | 18 | 12 | 6 |
Matrix representation of C92⋊4S3 ►in GL5(𝔽19)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 |
| 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 9 |
| 18 | 3 | 0 | 0 | 0 |
| 16 | 8 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 |
| 0 | 0 | 6 | 0 | 0 |
| 0 | 0 | 0 | 6 | 0 |
| 16 | 8 | 0 | 0 | 0 |
| 11 | 2 | 0 | 0 | 0 |
| 0 | 0 | 7 | 0 | 0 |
| 0 | 0 | 0 | 11 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 18 | 0 | 0 | 0 | 0 |
| 16 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 |
| 0 | 0 | 13 | 0 | 0 |
| 0 | 0 | 0 | 0 | 18 |
G:=sub<GL(5,GF(19))| [1,0,0,0,0,0,1,0,0,0,0,0,9,0,0,0,0,0,9,0,0,0,0,0,9],[18,16,0,0,0,3,8,0,0,0,0,0,0,6,0,0,0,0,0,6,0,0,9,0,0],[16,11,0,0,0,8,2,0,0,0,0,0,7,0,0,0,0,0,11,0,0,0,0,0,1],[18,16,0,0,0,0,1,0,0,0,0,0,0,13,0,0,0,3,0,0,0,0,0,0,18] >;
C92⋊4S3 in GAP, Magma, Sage, TeX
C_9^2\rtimes_4S_3
% in TeX
G:=Group("C9^2:4S3"); // GroupNames label
G:=SmallGroup(486,140);
// by ID
G=gap.SmallGroup(486,140);
# by ID
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,979,1190,338,867,735,3244]);
// Polycyclic
G:=Group<a,b,c,d|a^9=b^9=c^3=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=a^6*b,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations