direct product, metabelian, nilpotent (class 3), monomial, 3-elementary
Aliases: C2×C33⋊C9, C33⋊5C18, C34.8C6, C6.6C3≀C3, (C32×C6)⋊1C9, C32⋊C9⋊12C6, (C33×C6).1C3, (C3×C6).21He3, C6.3(C32⋊C9), C33.31(C3×C6), C32.7(C3×C18), C32.19(C2×He3), (C32×C6).19C32, (C3×C6).43- 1+2, C32.4(C2×3- 1+2), C3.1(C2×C3≀C3), (C3×C6).7(C3×C9), (C2×C32⋊C9)⋊4C3, C3.3(C2×C32⋊C9), SmallGroup(486,73)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C33⋊C9
G = < a,b,c,d,e | a2=b3=c3=d3=e9=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe-1=bc-1d, cd=dc, ece-1=cd-1, de=ed >
Subgroups: 504 in 180 conjugacy classes, 36 normal (14 characteristic)
C1, C2, C3, C3, C3, C6, C6, C6, C9, C32, C32, C32, C18, C3×C6, C3×C6, C3×C6, C3×C9, C33, C33, C33, C3×C18, C32×C6, C32×C6, C32×C6, C32⋊C9, C34, C2×C32⋊C9, C33×C6, C33⋊C9, C2×C33⋊C9
Quotients: C1, C2, C3, C6, C9, C32, C18, C3×C6, C3×C9, He3, 3- 1+2, C3×C18, C2×He3, C2×3- 1+2, C32⋊C9, C3≀C3, C2×C32⋊C9, C2×C3≀C3, C33⋊C9, C2×C33⋊C9
►On 54 points
Generators in S54
(1 54)(2 46)(3 47)(4 48)(5 49)(6 50)(7 51)(8 52)(9 53)(10 30)(11 31)(12 32)(13 33)(14 34)(15 35)(16 36)(17 28)(18 29)(19 39)(20 40)(21 41)(22 42)(23 43)(24 44)(25 45)(26 37)(27 38)
(1 41 28)(2 45 35)(3 30 43)(4 44 31)(5 39 29)(6 33 37)(7 38 34)(8 42 32)(9 36 40)(10 23 47)(11 48 24)(12 52 22)(13 26 50)(14 51 27)(15 46 25)(16 20 53)(17 54 21)(18 49 19)
(1 41 28)(2 8 5)(3 33 40)(4 44 31)(6 36 43)(7 38 34)(9 30 37)(10 26 53)(11 48 24)(12 18 15)(13 20 47)(14 51 27)(16 23 50)(17 54 21)(19 25 22)(29 35 32)(39 45 42)(46 52 49)
(1 44 34)(2 45 35)(3 37 36)(4 38 28)(5 39 29)(6 40 30)(7 41 31)(8 42 32)(9 43 33)(10 50 20)(11 51 21)(12 52 22)(13 53 23)(14 54 24)(15 46 25)(16 47 26)(17 48 27)(18 49 19)
(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)
G:=sub<Sym(54)| (1,54)(2,46)(3,47)(4,48)(5,49)(6,50)(7,51)(8,52)(9,53)(10,30)(11,31)(12,32)(13,33)(14,34)(15,35)(16,36)(17,28)(18,29)(19,39)(20,40)(21,41)(22,42)(23,43)(24,44)(25,45)(26,37)(27,38), (1,41,28)(2,45,35)(3,30,43)(4,44,31)(5,39,29)(6,33,37)(7,38,34)(8,42,32)(9,36,40)(10,23,47)(11,48,24)(12,52,22)(13,26,50)(14,51,27)(15,46,25)(16,20,53)(17,54,21)(18,49,19), (1,41,28)(2,8,5)(3,33,40)(4,44,31)(6,36,43)(7,38,34)(9,30,37)(10,26,53)(11,48,24)(12,18,15)(13,20,47)(14,51,27)(16,23,50)(17,54,21)(19,25,22)(29,35,32)(39,45,42)(46,52,49), (1,44,34)(2,45,35)(3,37,36)(4,38,28)(5,39,29)(6,40,30)(7,41,31)(8,42,32)(9,43,33)(10,50,20)(11,51,21)(12,52,22)(13,53,23)(14,54,24)(15,46,25)(16,47,26)(17,48,27)(18,49,19), (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)>;
G:=Group( (1,54)(2,46)(3,47)(4,48)(5,49)(6,50)(7,51)(8,52)(9,53)(10,30)(11,31)(12,32)(13,33)(14,34)(15,35)(16,36)(17,28)(18,29)(19,39)(20,40)(21,41)(22,42)(23,43)(24,44)(25,45)(26,37)(27,38), (1,41,28)(2,45,35)(3,30,43)(4,44,31)(5,39,29)(6,33,37)(7,38,34)(8,42,32)(9,36,40)(10,23,47)(11,48,24)(12,52,22)(13,26,50)(14,51,27)(15,46,25)(16,20,53)(17,54,21)(18,49,19), (1,41,28)(2,8,5)(3,33,40)(4,44,31)(6,36,43)(7,38,34)(9,30,37)(10,26,53)(11,48,24)(12,18,15)(13,20,47)(14,51,27)(16,23,50)(17,54,21)(19,25,22)(29,35,32)(39,45,42)(46,52,49), (1,44,34)(2,45,35)(3,37,36)(4,38,28)(5,39,29)(6,40,30)(7,41,31)(8,42,32)(9,43,33)(10,50,20)(11,51,21)(12,52,22)(13,53,23)(14,54,24)(15,46,25)(16,47,26)(17,48,27)(18,49,19), (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) );
G=PermutationGroup([[(1,54),(2,46),(3,47),(4,48),(5,49),(6,50),(7,51),(8,52),(9,53),(10,30),(11,31),(12,32),(13,33),(14,34),(15,35),(16,36),(17,28),(18,29),(19,39),(20,40),(21,41),(22,42),(23,43),(24,44),(25,45),(26,37),(27,38)], [(1,41,28),(2,45,35),(3,30,43),(4,44,31),(5,39,29),(6,33,37),(7,38,34),(8,42,32),(9,36,40),(10,23,47),(11,48,24),(12,52,22),(13,26,50),(14,51,27),(15,46,25),(16,20,53),(17,54,21),(18,49,19)], [(1,41,28),(2,8,5),(3,33,40),(4,44,31),(6,36,43),(7,38,34),(9,30,37),(10,26,53),(11,48,24),(12,18,15),(13,20,47),(14,51,27),(16,23,50),(17,54,21),(19,25,22),(29,35,32),(39,45,42),(46,52,49)], [(1,44,34),(2,45,35),(3,37,36),(4,38,28),(5,39,29),(6,40,30),(7,41,31),(8,42,32),(9,43,33),(10,50,20),(11,51,21),(12,52,22),(13,53,23),(14,54,24),(15,46,25),(16,47,26),(17,48,27),(18,49,19)], [(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)]])
102 conjugacy classes
| class | 1 | 2 | 3A | ··· | 3H | 3I | ··· | 3AF | 6A | ··· | 6H | 6I | ··· | 6AF | 9A | ··· | 9R | 18A | ··· | 18R |
| order | 1 | 2 | 3 | ··· | 3 | 3 | ··· | 3 | 6 | ··· | 6 | 6 | ··· | 6 | 9 | ··· | 9 | 18 | ··· | 18 |
| size | 1 | 1 | 1 | ··· | 1 | 3 | ··· | 3 | 1 | ··· | 1 | 3 | ··· | 3 | 9 | ··· | 9 | 9 | ··· | 9 |
102 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 3 | 3 |
| type | + | + | ||||||||||||
| image | C1 | C2 | C3 | C3 | C6 | C6 | C9 | C18 | He3 | 3- 1+2 | C2×He3 | C2×3- 1+2 | C3≀C3 | C2×C3≀C3 |
| kernel | C2×C33⋊C9 | C33⋊C9 | C2×C32⋊C9 | C33×C6 | C32⋊C9 | C34 | C32×C6 | C33 | C3×C6 | C3×C6 | C32 | C32 | C6 | C3 |
| # reps | 1 | 1 | 6 | 2 | 6 | 2 | 18 | 18 | 2 | 4 | 2 | 4 | 18 | 18 |
Matrix representation of C2×C33⋊C9 ►in GL4(𝔽19) generated by
| 18 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 11 | 0 | 0 | 0 |
| 0 | 11 | 0 | 0 |
| 0 | 0 | 7 | 0 |
| 0 | 0 | 0 | 7 |
| 1 | 0 | 0 | 0 |
| 0 | 11 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 7 |
| 1 | 0 | 0 | 0 |
| 0 | 7 | 0 | 0 |
| 0 | 0 | 7 | 0 |
| 0 | 0 | 0 | 7 |
| 9 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 11 | 0 | 0 |
G:=sub<GL(4,GF(19))| [18,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[11,0,0,0,0,11,0,0,0,0,7,0,0,0,0,7],[1,0,0,0,0,11,0,0,0,0,1,0,0,0,0,7],[1,0,0,0,0,7,0,0,0,0,7,0,0,0,0,7],[9,0,0,0,0,0,0,11,0,1,0,0,0,0,1,0] >;
C2×C33⋊C9 in GAP, Magma, Sage, TeX
C_2\times C_3^3\rtimes C_9
% in TeX
G:=Group("C2xC3^3:C9"); // GroupNames label
G:=SmallGroup(486,73);
// by ID
G=gap.SmallGroup(486,73);
# by ID
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,331,224,2169]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^3=c^3=d^3=e^9=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,e*b*e^-1=b*c^-1*d,c*d=d*c,e*c*e^-1=c*d^-1,d*e=e*d>;
// generators/relations