non-abelian, supersoluble, monomial
Aliases: C9⋊C9⋊2S3, C32⋊C9.7C6, (C32×C9).7S3, C32.6(C9⋊C6), C33.42(C3×S3), C32⋊2D9.6C3, C33.31C32⋊C2, C3.3(He3.4C6), C3.9(C33.S3), (C3×C9).5(C3×S3), (C3×C9).5(C3⋊S3), C32.42(C3×C3⋊S3), SmallGroup(486,152)
Series: Derived ►Chief ►Lower central ►Upper central
| C32⋊C9 — C9⋊C9⋊2S3 |
Generators and relations for C9⋊C9⋊2S3
G = < a,b,c,d | a9=b9=c3=d2=1, bab-1=a7, cac-1=ab3, dad=a-1b6, bc=cb, bd=db, dcd=c-1 >
Subgroups: 326 in 72 conjugacy classes, 20 normal (11 characteristic)
C1, C2, C3, C3, S3, C6, C9, C32, C32, C32, D9, C18, C3×S3, C3⋊S3, C3×C9, C3×C9, C3×C9, C33, C3×D9, S3×C9, C3×C3⋊S3, C32⋊C9, C32⋊C9, C9⋊C9, C9⋊C9, C32×C9, C9⋊C18, C32⋊2D9, C9×C3⋊S3, C33.31C32, C9⋊C9⋊2S3
Quotients: C1, C2, C3, S3, C6, C3×S3, C3⋊S3, C9⋊C6, C3×C3⋊S3, C33.S3, He3.4C6, C9⋊C9⋊2S3
►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 35 43 4 29 37 7 32 40)(2 30 41 5 33 44 8 36 38)(3 34 39 6 28 42 9 31 45)(10 27 48 16 24 54 13 21 51)(11 22 46 17 19 52 14 25 49)(12 26 53 18 23 50 15 20 47)
(1 41 28)(2 39 32)(3 37 36)(4 44 31)(5 42 35)(6 40 30)(7 38 34)(8 45 29)(9 43 33)(10 19 47)(11 23 54)(12 27 52)(13 22 50)(14 26 48)(15 21 46)(16 25 53)(17 20 51)(18 24 49)
(1 20)(2 22)(3 24)(4 26)(5 19)(6 21)(7 23)(8 25)(9 27)(10 42)(11 38)(12 43)(13 39)(14 44)(15 40)(16 45)(17 41)(18 37)(28 51)(29 53)(30 46)(31 48)(32 50)(33 52)(34 54)(35 47)(36 49)
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), (1,35,43,4,29,37,7,32,40)(2,30,41,5,33,44,8,36,38)(3,34,39,6,28,42,9,31,45)(10,27,48,16,24,54,13,21,51)(11,22,46,17,19,52,14,25,49)(12,26,53,18,23,50,15,20,47), (1,41,28)(2,39,32)(3,37,36)(4,44,31)(5,42,35)(6,40,30)(7,38,34)(8,45,29)(9,43,33)(10,19,47)(11,23,54)(12,27,52)(13,22,50)(14,26,48)(15,21,46)(16,25,53)(17,20,51)(18,24,49), (1,20)(2,22)(3,24)(4,26)(5,19)(6,21)(7,23)(8,25)(9,27)(10,42)(11,38)(12,43)(13,39)(14,44)(15,40)(16,45)(17,41)(18,37)(28,51)(29,53)(30,46)(31,48)(32,50)(33,52)(34,54)(35,47)(36,49)>;
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), (1,35,43,4,29,37,7,32,40)(2,30,41,5,33,44,8,36,38)(3,34,39,6,28,42,9,31,45)(10,27,48,16,24,54,13,21,51)(11,22,46,17,19,52,14,25,49)(12,26,53,18,23,50,15,20,47), (1,41,28)(2,39,32)(3,37,36)(4,44,31)(5,42,35)(6,40,30)(7,38,34)(8,45,29)(9,43,33)(10,19,47)(11,23,54)(12,27,52)(13,22,50)(14,26,48)(15,21,46)(16,25,53)(17,20,51)(18,24,49), (1,20)(2,22)(3,24)(4,26)(5,19)(6,21)(7,23)(8,25)(9,27)(10,42)(11,38)(12,43)(13,39)(14,44)(15,40)(16,45)(17,41)(18,37)(28,51)(29,53)(30,46)(31,48)(32,50)(33,52)(34,54)(35,47)(36,49) );
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)], [(1,35,43,4,29,37,7,32,40),(2,30,41,5,33,44,8,36,38),(3,34,39,6,28,42,9,31,45),(10,27,48,16,24,54,13,21,51),(11,22,46,17,19,52,14,25,49),(12,26,53,18,23,50,15,20,47)], [(1,41,28),(2,39,32),(3,37,36),(4,44,31),(5,42,35),(6,40,30),(7,38,34),(8,45,29),(9,43,33),(10,19,47),(11,23,54),(12,27,52),(13,22,50),(14,26,48),(15,21,46),(16,25,53),(17,20,51),(18,24,49)], [(1,20),(2,22),(3,24),(4,26),(5,19),(6,21),(7,23),(8,25),(9,27),(10,42),(11,38),(12,43),(13,39),(14,44),(15,40),(16,45),(17,41),(18,37),(28,51),(29,53),(30,46),(31,48),(32,50),(33,52),(34,54),(35,47),(36,49)]])
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 |
| type | + | + | + | + | + | ||||||
| image | C1 | C2 | C3 | C6 | S3 | S3 | C3×S3 | C3×S3 | He3.4C6 | C9⋊C6 | C9⋊C9⋊2S3 |
| kernel | C9⋊C9⋊2S3 | C33.31C32 | C32⋊2D9 | C32⋊C9 | C9⋊C9 | C32×C9 | C3×C9 | C33 | C3 | C32 | C1 |
| # reps | 1 | 1 | 2 | 2 | 3 | 1 | 6 | 2 | 12 | 3 | 6 |
Matrix representation of C9⋊C9⋊2S3 ►in GL6(𝔽19)
| 5 | 0 | 0 | 0 | 0 | 0 |
| 0 | 17 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 6 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 0 | 17 | 0 | 0 | 0 | 0 |
| 0 | 0 | 17 | 0 | 0 | 0 |
| 17 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 17 | 0 |
| 0 | 0 | 0 | 0 | 0 | 17 |
| 0 | 0 | 0 | 17 | 0 | 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))| [5,0,0,0,0,0,0,17,0,0,0,0,0,0,16,0,0,0,0,0,0,9,0,0,0,0,0,0,6,0,0,0,0,0,0,4],[0,0,17,0,0,0,17,0,0,0,0,0,0,17,0,0,0,0,0,0,0,0,0,17,0,0,0,17,0,0,0,0,0,0,17,0],[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] >;
C9⋊C9⋊2S3 in GAP, Magma, Sage, TeX
C_9\rtimes C_9\rtimes_2S_3
% in TeX
G:=Group("C9:C9:2S3"); // GroupNames label
G:=SmallGroup(486,152);
// by ID
G=gap.SmallGroup(486,152);
# by ID
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,979,1190,548,338,867,735,3244]);
// Polycyclic
G:=Group<a,b,c,d|a^9=b^9=c^3=d^2=1,b*a*b^-1=a^7,c*a*c^-1=a*b^3,d*a*d=a^-1*b^6,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations