direct product, metabelian, nilpotent (class 4), monomial, 3-elementary
Aliases: C2xC92:C3, C92:14C6, (C9xC18):1C3, (C3xC6).1He3, C3.He3:2C6, C32.1(C2xHe3), He3:C3.3C6, (C3xC18).16C32, C6.6(He3:C3), (C3xC9).17(C3xC6), (C2xC3.He3):1C3, C3.6(C2xHe3:C3), (C2xHe3:C3).1C3, SmallGroup(486,85)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2xC92:C3
G = < a,b,c,d | a2=b9=c9=d3=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b7c2, dcd-1=b3c7 >
Quotients: C1, C2, C3, C6, C32, C3xC6, He3, C2xHe3, He3:C3, C2xHe3:C3, C92:C3, C2xC92:C3
Smallest permutation representation of C2xC92:C3
►On 54 points
Generators in S54
(1 18)(2 16)(3 17)(4 10)(5 11)(6 12)(7 14)(8 15)(9 13)(19 50)(20 51)(21 52)(22 53)(23 54)(24 46)(25 47)(26 48)(27 49)(28 43)(29 44)(30 45)(31 37)(32 38)(33 39)(34 40)(35 41)(36 42)
(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 8 5 2 9 6 3 7 4)(10 18 15 11 16 13 12 17 14)(19 24 20 25 21 26 22 27 23)(28 35 33 31 29 36 34 32 30)(37 44 42 40 38 45 43 41 39)(46 51 47 52 48 53 49 54 50)
(1 24 29)(2 21 32)(3 27 35)(4 22 31)(5 19 34)(6 25 28)(7 20 33)(8 26 36)(9 23 30)(10 53 37)(11 50 40)(12 47 43)(13 54 45)(14 51 39)(15 48 42)(16 52 38)(17 49 41)(18 46 44)
G:=sub<Sym(54)| (1,18)(2,16)(3,17)(4,10)(5,11)(6,12)(7,14)(8,15)(9,13)(19,50)(20,51)(21,52)(22,53)(23,54)(24,46)(25,47)(26,48)(27,49)(28,43)(29,44)(30,45)(31,37)(32,38)(33,39)(34,40)(35,41)(36,42), (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), (1,8,5,2,9,6,3,7,4)(10,18,15,11,16,13,12,17,14)(19,24,20,25,21,26,22,27,23)(28,35,33,31,29,36,34,32,30)(37,44,42,40,38,45,43,41,39)(46,51,47,52,48,53,49,54,50), (1,24,29)(2,21,32)(3,27,35)(4,22,31)(5,19,34)(6,25,28)(7,20,33)(8,26,36)(9,23,30)(10,53,37)(11,50,40)(12,47,43)(13,54,45)(14,51,39)(15,48,42)(16,52,38)(17,49,41)(18,46,44)>;
G:=Group( (1,18)(2,16)(3,17)(4,10)(5,11)(6,12)(7,14)(8,15)(9,13)(19,50)(20,51)(21,52)(22,53)(23,54)(24,46)(25,47)(26,48)(27,49)(28,43)(29,44)(30,45)(31,37)(32,38)(33,39)(34,40)(35,41)(36,42), (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), (1,8,5,2,9,6,3,7,4)(10,18,15,11,16,13,12,17,14)(19,24,20,25,21,26,22,27,23)(28,35,33,31,29,36,34,32,30)(37,44,42,40,38,45,43,41,39)(46,51,47,52,48,53,49,54,50), (1,24,29)(2,21,32)(3,27,35)(4,22,31)(5,19,34)(6,25,28)(7,20,33)(8,26,36)(9,23,30)(10,53,37)(11,50,40)(12,47,43)(13,54,45)(14,51,39)(15,48,42)(16,52,38)(17,49,41)(18,46,44) );
G=PermutationGroup([[(1,18),(2,16),(3,17),(4,10),(5,11),(6,12),(7,14),(8,15),(9,13),(19,50),(20,51),(21,52),(22,53),(23,54),(24,46),(25,47),(26,48),(27,49),(28,43),(29,44),(30,45),(31,37),(32,38),(33,39),(34,40),(35,41),(36,42)], [(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)], [(1,8,5,2,9,6,3,7,4),(10,18,15,11,16,13,12,17,14),(19,24,20,25,21,26,22,27,23),(28,35,33,31,29,36,34,32,30),(37,44,42,40,38,45,43,41,39),(46,51,47,52,48,53,49,54,50)], [(1,24,29),(2,21,32),(3,27,35),(4,22,31),(5,19,34),(6,25,28),(7,20,33),(8,26,36),(9,23,30),(10,53,37),(11,50,40),(12,47,43),(13,54,45),(14,51,39),(15,48,42),(16,52,38),(17,49,41),(18,46,44)]])
70 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 3F | 6A | 6B | 6C | 6D | 6E | 6F | 9A | ··· | 9X | 9Y | 9Z | 9AA | 9AB | 18A | ··· | 18X | 18Y | 18Z | 18AA | 18AB |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | 6 | 6 | 6 | 6 | 6 | 6 | 9 | ··· | 9 | 9 | 9 | 9 | 9 | 18 | ··· | 18 | 18 | 18 | 18 | 18 |
| size | 1 | 1 | 1 | 1 | 3 | 3 | 27 | 27 | 1 | 1 | 3 | 3 | 27 | 27 | 3 | ··· | 3 | 27 | 27 | 27 | 27 | 3 | ··· | 3 | 27 | 27 | 27 | 27 |
70 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 3 | 3 |
| type | + | + | ||||||||||||
| image | C1 | C2 | C3 | C3 | C3 | C6 | C6 | C6 | He3 | C2xHe3 | He3:C3 | C2xHe3:C3 | C92:C3 | C2xC92:C3 |
| kernel | C2xC92:C3 | C92:C3 | C9xC18 | C2xHe3:C3 | C2xC3.He3 | C92 | He3:C3 | C3.He3 | C3xC6 | C32 | C6 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 4 | 2 | 2 | 4 | 2 | 2 | 6 | 6 | 18 | 18 |
Matrix representation of C2xC92:C3 ►in GL3(F19) generated by
| 18 | 0 | 0 |
| 0 | 18 | 0 |
| 0 | 0 | 18 |
| 7 | 0 | 0 |
| 0 | 4 | 0 |
| 0 | 0 | 5 |
| 6 | 0 | 0 |
| 0 | 6 | 0 |
| 0 | 0 | 9 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
G:=sub<GL(3,GF(19))| [18,0,0,0,18,0,0,0,18],[7,0,0,0,4,0,0,0,5],[6,0,0,0,6,0,0,0,9],[0,0,1,1,0,0,0,1,0] >;
C2xC92:C3 in GAP, Magma, Sage, TeX
C_2\times C_9^2\rtimes C_3
% in TeX
G:=Group("C2xC9^2:C3"); // GroupNames label
G:=SmallGroup(486,85);
// by ID
G=gap.SmallGroup(486,85);
# by ID
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,224,338,873,453,3250]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^9=c^9=d^3=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^7*c^2,d*c*d^-1=b^3*c^7>;
// generators/relations
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