direct product, metabelian, nilpotent (class 4), monomial, 3-elementary
Aliases: C2xC92:2C3, C92:15C6, (C9xC18):2C3, (C3xC6).2He3, He3:C3:5C6, C32.2(C2xHe3), (C3xC18).17C32, C6.7(He3:C3), (C3xC9).18(C3xC6), (C2xHe3:C3):1C3, C3.7(C2xHe3:C3), SmallGroup(486,86)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2xC92:2C3
G = < a,b,c,d | a2=b9=c9=d3=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b7c-1, dcd-1=b3c >
Quotients: C1, C2, C3, C6, C32, C3xC6, He3, C2xHe3, He3:C3, C2xHe3:C3, C92:2C3, C2xC92:2C3
3C3
27C3
27C3
27C3
3C6
27C6
27C6
27C6
3C9
3C9
3C9
3C9
9C32
9C32
9C32
3C18
3C18
3C18
3C18
3He3
3He3
3He3
Smallest permutation representation of C2xC92:2C3
►On 54 points
Generators in S54
(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)(13 16)(14 17)(15 18)(19 54)(20 46)(21 47)(22 48)(23 49)(24 50)(25 51)(26 52)(27 53)(28 38)(29 39)(30 40)(31 41)(32 42)(33 43)(34 44)(35 45)(36 37)
(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 18 4 3 17 6 2 16 5)(7 15 10 9 14 12 8 13 11)(19 20 21 22 23 24 25 26 27)(28 30 32 34 36 29 31 33 35)(37 39 41 43 45 38 40 42 44)(46 47 48 49 50 51 52 53 54)
(1 36 47)(2 30 53)(3 33 50)(4 31 46)(5 34 52)(6 28 49)(7 37 21)(8 40 27)(9 43 24)(10 41 20)(11 44 26)(12 38 23)(13 42 22)(14 45 19)(15 39 25)(16 32 48)(17 35 54)(18 29 51)
G:=sub<Sym(54)| (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,16)(14,17)(15,18)(19,54)(20,46)(21,47)(22,48)(23,49)(24,50)(25,51)(26,52)(27,53)(28,38)(29,39)(30,40)(31,41)(32,42)(33,43)(34,44)(35,45)(36,37), (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), (1,18,4,3,17,6,2,16,5)(7,15,10,9,14,12,8,13,11)(19,20,21,22,23,24,25,26,27)(28,30,32,34,36,29,31,33,35)(37,39,41,43,45,38,40,42,44)(46,47,48,49,50,51,52,53,54), (1,36,47)(2,30,53)(3,33,50)(4,31,46)(5,34,52)(6,28,49)(7,37,21)(8,40,27)(9,43,24)(10,41,20)(11,44,26)(12,38,23)(13,42,22)(14,45,19)(15,39,25)(16,32,48)(17,35,54)(18,29,51)>;
G:=Group( (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,16)(14,17)(15,18)(19,54)(20,46)(21,47)(22,48)(23,49)(24,50)(25,51)(26,52)(27,53)(28,38)(29,39)(30,40)(31,41)(32,42)(33,43)(34,44)(35,45)(36,37), (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), (1,18,4,3,17,6,2,16,5)(7,15,10,9,14,12,8,13,11)(19,20,21,22,23,24,25,26,27)(28,30,32,34,36,29,31,33,35)(37,39,41,43,45,38,40,42,44)(46,47,48,49,50,51,52,53,54), (1,36,47)(2,30,53)(3,33,50)(4,31,46)(5,34,52)(6,28,49)(7,37,21)(8,40,27)(9,43,24)(10,41,20)(11,44,26)(12,38,23)(13,42,22)(14,45,19)(15,39,25)(16,32,48)(17,35,54)(18,29,51) );
G=PermutationGroup([[(1,7),(2,8),(3,9),(4,10),(5,11),(6,12),(13,16),(14,17),(15,18),(19,54),(20,46),(21,47),(22,48),(23,49),(24,50),(25,51),(26,52),(27,53),(28,38),(29,39),(30,40),(31,41),(32,42),(33,43),(34,44),(35,45),(36,37)], [(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)], [(1,18,4,3,17,6,2,16,5),(7,15,10,9,14,12,8,13,11),(19,20,21,22,23,24,25,26,27),(28,30,32,34,36,29,31,33,35),(37,39,41,43,45,38,40,42,44),(46,47,48,49,50,51,52,53,54)], [(1,36,47),(2,30,53),(3,33,50),(4,31,46),(5,34,52),(6,28,49),(7,37,21),(8,40,27),(9,43,24),(10,41,20),(11,44,26),(12,38,23),(13,42,22),(14,45,19),(15,39,25),(16,32,48),(17,35,54),(18,29,51)]])
70 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | ··· | 3J | 6A | 6B | 6C | 6D | 6E | ··· | 6J | 9A | ··· | 9X | 18A | ··· | 18X |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 3 | ··· | 3 | 6 | 6 | 6 | 6 | 6 | ··· | 6 | 9 | ··· | 9 | 18 | ··· | 18 |
| size | 1 | 1 | 1 | 1 | 3 | 3 | 27 | ··· | 27 | 1 | 1 | 3 | 3 | 27 | ··· | 27 | 3 | ··· | 3 | 3 | ··· | 3 |
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 | C2xHe3 | He3:C3 | C2xHe3:C3 | C92:2C3 | C2xC92:2C3 |
| kernel | C2xC92:2C3 | C92:2C3 | C9xC18 | C2xHe3:C3 | C92 | He3:C3 | C3xC6 | C32 | C6 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 6 | 2 | 6 | 2 | 2 | 6 | 6 | 18 | 18 |
Matrix representation of C2xC92:2C3 ►in GL3(F19) generated by
| 18 | 0 | 0 |
| 0 | 18 | 0 |
| 0 | 0 | 18 |
| 11 | 0 | 0 |
| 0 | 4 | 0 |
| 0 | 0 | 16 |
| 9 | 0 | 0 |
| 0 | 4 | 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],[11,0,0,0,4,0,0,0,16],[9,0,0,0,4,0,0,0,9],[0,0,1,1,0,0,0,1,0] >;
C2xC92:2C3 in GAP, Magma, Sage, TeX
C_2\times C_9^2\rtimes_2C_3
% in TeX
G:=Group("C2xC9^2:2C3"); // GroupNames label
G:=SmallGroup(486,86);
// by ID
G=gap.SmallGroup(486,86);
# by ID
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,224,824,873,453,3250]);
// 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^-1,d*c*d^-1=b^3*c>;
// generators/relations
Export