non-abelian, supersoluble, monomial
Aliases: C92⋊5S3, C9⋊C9⋊4S3, C92⋊5C3⋊1C2, C32⋊C9.9C6, C32⋊C9.19S3, C33.23(C3×S3), C32⋊2D9.8C3, C3.5(He3.4C6), C3.13(He3.4S3), (C3×C9).7(C3×S3), (C3×C9).7(C3⋊S3), C32.46(C3×C3⋊S3), SmallGroup(486,156)
Series: Derived ►Chief ►Lower central ►Upper central
| C32⋊C9 — C92⋊5S3 |
Generators and relations for C92⋊5S3
G = < a,b,c,d | a9=b9=c3=d2=1, ab=ba, cac-1=ab3, ad=da, cbc-1=a6b, dbd=b-1, dcd=c-1 >
Subgroups: 290 in 58 conjugacy classes, 17 normal (13 characteristic)
C1, C2, C3, C3, S3, C6, C9, 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, C9⋊C9, C9×D9, C32⋊C18, C9⋊C18, C32⋊2D9, C92⋊5C3, C92⋊5S3
Quotients: C1, C2, C3, S3, C6, C3×S3, C3⋊S3, C3×C3⋊S3, He3.4S3, He3.4C6, C92⋊5S3
►On 54 points
Generators in S54
(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 28 40 7 34 37 4 31 43)(2 29 41 8 35 38 5 32 44)(3 30 42 9 36 39 6 33 45)(10 50 19 13 53 22 16 47 25)(11 51 20 14 54 23 17 48 26)(12 52 21 15 46 24 18 49 27)
(1 29 39)(2 33 37)(3 28 44)(4 32 42)(5 36 40)(6 31 38)(7 35 45)(8 30 43)(9 34 41)(10 52 20)(11 50 24)(12 48 19)(13 46 23)(14 53 27)(15 51 22)(16 49 26)(17 47 21)(18 54 25)
(1 15)(2 16)(3 17)(4 18)(5 10)(6 11)(7 12)(8 13)(9 14)(19 35)(20 36)(21 28)(22 29)(23 30)(24 31)(25 32)(26 33)(27 34)(37 49)(38 50)(39 51)(40 52)(41 53)(42 54)(43 46)(44 47)(45 48)
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), (1,28,40,7,34,37,4,31,43)(2,29,41,8,35,38,5,32,44)(3,30,42,9,36,39,6,33,45)(10,50,19,13,53,22,16,47,25)(11,51,20,14,54,23,17,48,26)(12,52,21,15,46,24,18,49,27), (1,29,39)(2,33,37)(3,28,44)(4,32,42)(5,36,40)(6,31,38)(7,35,45)(8,30,43)(9,34,41)(10,52,20)(11,50,24)(12,48,19)(13,46,23)(14,53,27)(15,51,22)(16,49,26)(17,47,21)(18,54,25), (1,15)(2,16)(3,17)(4,18)(5,10)(6,11)(7,12)(8,13)(9,14)(19,35)(20,36)(21,28)(22,29)(23,30)(24,31)(25,32)(26,33)(27,34)(37,49)(38,50)(39,51)(40,52)(41,53)(42,54)(43,46)(44,47)(45,48)>;
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), (1,28,40,7,34,37,4,31,43)(2,29,41,8,35,38,5,32,44)(3,30,42,9,36,39,6,33,45)(10,50,19,13,53,22,16,47,25)(11,51,20,14,54,23,17,48,26)(12,52,21,15,46,24,18,49,27), (1,29,39)(2,33,37)(3,28,44)(4,32,42)(5,36,40)(6,31,38)(7,35,45)(8,30,43)(9,34,41)(10,52,20)(11,50,24)(12,48,19)(13,46,23)(14,53,27)(15,51,22)(16,49,26)(17,47,21)(18,54,25), (1,15)(2,16)(3,17)(4,18)(5,10)(6,11)(7,12)(8,13)(9,14)(19,35)(20,36)(21,28)(22,29)(23,30)(24,31)(25,32)(26,33)(27,34)(37,49)(38,50)(39,51)(40,52)(41,53)(42,54)(43,46)(44,47)(45,48) );
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)], [(1,28,40,7,34,37,4,31,43),(2,29,41,8,35,38,5,32,44),(3,30,42,9,36,39,6,33,45),(10,50,19,13,53,22,16,47,25),(11,51,20,14,54,23,17,48,26),(12,52,21,15,46,24,18,49,27)], [(1,29,39),(2,33,37),(3,28,44),(4,32,42),(5,36,40),(6,31,38),(7,35,45),(8,30,43),(9,34,41),(10,52,20),(11,50,24),(12,48,19),(13,46,23),(14,53,27),(15,51,22),(16,49,26),(17,47,21),(18,54,25)], [(1,15),(2,16),(3,17),(4,18),(5,10),(6,11),(7,12),(8,13),(9,14),(19,35),(20,36),(21,28),(22,29),(23,30),(24,31),(25,32),(26,33),(27,34),(37,49),(38,50),(39,51),(40,52),(41,53),(42,54),(43,46),(44,47),(45,48)]])
39 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 3F | 6A | 6B | 9A | ··· | 9F | 9G | ··· | 9O | 9P | ··· | 9W | 18A | ··· | 18F |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | 6 | 6 | 9 | ··· | 9 | 9 | ··· | 9 | 9 | ··· | 9 | 18 | ··· | 18 |
| size | 1 | 27 | 1 | 1 | 2 | 2 | 2 | 18 | 27 | 27 | 3 | ··· | 3 | 6 | ··· | 6 | 18 | ··· | 18 | 27 | ··· | 27 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 6 | 6 |
| type | + | + | + | + | + | + | ||||||
| image | C1 | C2 | C3 | C6 | S3 | S3 | S3 | C3×S3 | C3×S3 | He3.4C6 | He3.4S3 | C92⋊5S3 |
| kernel | C92⋊5S3 | C92⋊5C3 | C32⋊2D9 | C32⋊C9 | C92 | C32⋊C9 | C9⋊C9 | C3×C9 | C33 | C3 | C3 | C1 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 6 | 2 | 12 | 3 | 6 |
Matrix representation of C92⋊5S3 ►in GL6(𝔽19)
| 0 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 17 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 0 | 0 | 0 | 17 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 | 0 |
| 6 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 0 | 0 | 0 | 5 | 0 | 0 |
| 0 | 0 | 0 | 0 | 17 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
G:=sub<GL(6,GF(19))| [0,0,17,0,0,0,5,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,17,0,0,0,5,0,0,0,0,0,0,16,0],[0,0,6,0,0,0,4,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,17,0,0,0,16,0,0],[0,1,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0],[0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0] >;
C92⋊5S3 in GAP, Magma, Sage, TeX
C_9^2\rtimes_5S_3
% in TeX
G:=Group("C9^2:5S3"); // GroupNames label
G:=SmallGroup(486,156);
// by ID
G=gap.SmallGroup(486,156);
# by ID
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,979,1190,338,867,873,735,3244]);
// Polycyclic
G:=Group<a,b,c,d|a^9=b^9=c^3=d^2=1,a*b=b*a,c*a*c^-1=a*b^3,a*d=d*a,c*b*c^-1=a^6*b,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations