direct product, metabelian, nilpotent (class 4), monomial, 3-elementary
Aliases: C2×C92.C3, C92.3C6, (C9×C18).2C3, (C3×C6).3He3, C32.3(C2×He3), (C3×C18).18C32, C3.He3.4C6, C6.8(He3⋊C3), (C3×C9).19(C3×C6), C3.8(C2×He3⋊C3), (C2×C3.He3).1C3, SmallGroup(486,87)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C92.C3
G = < a,b,c,d | a2=b9=c9=1, d3=c6, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b7c-1, dcd-1=b3c >
Smallest permutation representation of C2×C92.C3
►On 54 points
Generators in S54
(1 14)(2 15)(3 13)(4 17)(5 18)(6 16)(7 10)(8 11)(9 12)(19 46)(20 47)(21 48)(22 49)(23 50)(24 51)(25 52)(26 53)(27 54)(28 44)(29 45)(30 37)(31 38)(32 39)(33 40)(34 41)(35 42)(36 43)
(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 12 18 3 11 17 2 10 16)(4 15 7 6 14 9 5 13 8)(19 21 23 25 27 20 22 24 26)(28 29 30 31 32 33 34 35 36)(37 38 39 40 41 42 43 44 45)(46 48 50 52 54 47 49 51 53)
(1 52 38 2 46 44 3 49 41)(4 26 33 5 20 30 6 23 36)(7 21 32 8 24 29 9 27 35)(10 48 39 11 51 45 12 54 42)(13 22 34 14 25 31 15 19 28)(16 50 43 17 53 40 18 47 37)
G:=sub<Sym(54)| (1,14)(2,15)(3,13)(4,17)(5,18)(6,16)(7,10)(8,11)(9,12)(19,46)(20,47)(21,48)(22,49)(23,50)(24,51)(25,52)(26,53)(27,54)(28,44)(29,45)(30,37)(31,38)(32,39)(33,40)(34,41)(35,42)(36,43), (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), (1,12,18,3,11,17,2,10,16)(4,15,7,6,14,9,5,13,8)(19,21,23,25,27,20,22,24,26)(28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45)(46,48,50,52,54,47,49,51,53), (1,52,38,2,46,44,3,49,41)(4,26,33,5,20,30,6,23,36)(7,21,32,8,24,29,9,27,35)(10,48,39,11,51,45,12,54,42)(13,22,34,14,25,31,15,19,28)(16,50,43,17,53,40,18,47,37)>;
G:=Group( (1,14)(2,15)(3,13)(4,17)(5,18)(6,16)(7,10)(8,11)(9,12)(19,46)(20,47)(21,48)(22,49)(23,50)(24,51)(25,52)(26,53)(27,54)(28,44)(29,45)(30,37)(31,38)(32,39)(33,40)(34,41)(35,42)(36,43), (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), (1,12,18,3,11,17,2,10,16)(4,15,7,6,14,9,5,13,8)(19,21,23,25,27,20,22,24,26)(28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45)(46,48,50,52,54,47,49,51,53), (1,52,38,2,46,44,3,49,41)(4,26,33,5,20,30,6,23,36)(7,21,32,8,24,29,9,27,35)(10,48,39,11,51,45,12,54,42)(13,22,34,14,25,31,15,19,28)(16,50,43,17,53,40,18,47,37) );
G=PermutationGroup([[(1,14),(2,15),(3,13),(4,17),(5,18),(6,16),(7,10),(8,11),(9,12),(19,46),(20,47),(21,48),(22,49),(23,50),(24,51),(25,52),(26,53),(27,54),(28,44),(29,45),(30,37),(31,38),(32,39),(33,40),(34,41),(35,42),(36,43)], [(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)], [(1,12,18,3,11,17,2,10,16),(4,15,7,6,14,9,5,13,8),(19,21,23,25,27,20,22,24,26),(28,29,30,31,32,33,34,35,36),(37,38,39,40,41,42,43,44,45),(46,48,50,52,54,47,49,51,53)], [(1,52,38,2,46,44,3,49,41),(4,26,33,5,20,30,6,23,36),(7,21,32,8,24,29,9,27,35),(10,48,39,11,51,45,12,54,42),(13,22,34,14,25,31,15,19,28),(16,50,43,17,53,40,18,47,37)]])
70 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 6A | 6B | 6C | 6D | 9A | ··· | 9X | 9Y | ··· | 9AD | 18A | ··· | 18X | 18Y | ··· | 18AD |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 6 | 6 | 6 | 6 | 9 | ··· | 9 | 9 | ··· | 9 | 18 | ··· | 18 | 18 | ··· | 18 |
| size | 1 | 1 | 1 | 1 | 3 | 3 | 1 | 1 | 3 | 3 | 3 | ··· | 3 | 27 | ··· | 27 | 3 | ··· | 3 | 27 | ··· | 27 |
70 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 3 | 3 |
| type | + | + | ||||||||||
| image | C1 | C2 | C3 | C3 | C6 | C6 | He3 | C2×He3 | He3⋊C3 | C2×He3⋊C3 | C92.C3 | C2×C92.C3 |
| kernel | C2×C92.C3 | C92.C3 | C9×C18 | C2×C3.He3 | C92 | C3.He3 | C3×C6 | C32 | C6 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 6 | 2 | 6 | 2 | 2 | 6 | 6 | 18 | 18 |
Matrix representation of C2×C92.C3 ►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 | 4 | 0 |
| 0 | 0 | 0 | 16 |
| 1 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 9 |
| 11 | 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],[11,0,0,0,0,11,0,0,0,0,4,0,0,0,0,16],[1,0,0,0,0,9,0,0,0,0,4,0,0,0,0,9],[11,0,0,0,0,0,0,11,0,1,0,0,0,0,1,0] >;
C2×C92.C3 in GAP, Magma, Sage, TeX
C_2\times C_9^2.C_3
% in TeX
G:=Group("C2xC9^2.C3"); // GroupNames label
G:=SmallGroup(486,87);
// by ID
G=gap.SmallGroup(486,87);
# by ID
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,979,224,824,873,453,3250]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^9=c^9=1,d^3=c^6,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^7*c^-1,d*c*d^-1=b^3*c>;
// generators/relations
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