direct product, metabelian, nilpotent (class 3), monomial, 3-elementary
Aliases: C2xC9.He3, C18.6He3, C3wrC3:9C6, C9oHe3:5C6, C9.6(C2xHe3), (C32xC18):8C3, (C32xC9):34C6, He3.9(C3xC6), (C3xC6).5C33, C6.11(C3xHe3), C3.11(C6xHe3), He3.C3:11C6, C3.He3:8C6, C33.46(C3xC6), He3:C3:13C6, (C3xC18).22C32, (C2xHe3).2C32, C32.5(C32xC6), (C32xC6).34C32, 3- 1+2.2(C3xC6), (C2x3- 1+2).2C32, (C2xC3wrC3):3C3, (C3xC9).31(C3xC6), (C2xC9oHe3):1C3, (C2xHe3.C3):3C3, (C2xHe3:C3):5C3, (C2xC3.He3):6C3, SmallGroup(486,214)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2xC9.He3
G = < a,b,c,d,e | a2=b9=c3=d3=e3=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe-1=b4, cd=dc, ece-1=b3cd-1, ede-1=b6d >
Subgroups: 306 in 134 conjugacy classes, 66 normal (22 characteristic)
C1, C2, C3, C3, C6, C6, C9, C9, C9, C32, C32, C18, C18, C18, C3xC6, C3xC6, C3xC9, C3xC9, He3, 3- 1+2, 3- 1+2, C33, C3xC18, C3xC18, C2xHe3, C2x3- 1+2, C2x3- 1+2, C32xC6, C3wrC3, He3.C3, He3:C3, C3.He3, C32xC9, C9oHe3, C2xC3wrC3, C2xHe3.C3, C2xHe3:C3, C2xC3.He3, C32xC18, C2xC9oHe3, C9.He3, C2xC9.He3
Quotients: C1, C2, C3, C6, C32, C3xC6, He3, C33, C2xHe3, C32xC6, C3xHe3, C6xHe3, C9.He3, C2xC9.He3
►On 54 points
(1 35)(2 36)(3 28)(4 29)(5 30)(6 31)(7 32)(8 33)(9 34)(10 37)(11 38)(12 39)(13 40)(14 41)(15 42)(16 43)(17 44)(18 45)(19 46)(20 47)(21 48)(22 49)(23 50)(24 51)(25 52)(26 53)(27 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 7 4)(2 8 5)(3 9 6)(10 16 13)(11 17 14)(12 18 15)(28 34 31)(29 35 32)(30 36 33)(37 43 40)(38 44 41)(39 45 42)
(10 13 16)(11 14 17)(12 15 18)(19 25 22)(20 26 23)(21 27 24)(37 40 43)(38 41 44)(39 42 45)(46 52 49)(47 53 50)(48 54 51)
(1 20 14)(2 27 18)(3 25 13)(4 23 17)(5 21 12)(6 19 16)(7 26 11)(8 24 15)(9 22 10)(28 52 40)(29 50 44)(30 48 39)(31 46 43)(32 53 38)(33 51 42)(34 49 37)(35 47 41)(36 54 45)
G:=sub<Sym(54)| (1,35)(2,36)(3,28)(4,29)(5,30)(6,31)(7,32)(8,33)(9,34)(10,37)(11,38)(12,39)(13,40)(14,41)(15,42)(16,43)(17,44)(18,45)(19,46)(20,47)(21,48)(22,49)(23,50)(24,51)(25,52)(26,53)(27,54), (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), (1,7,4)(2,8,5)(3,9,6)(10,16,13)(11,17,14)(12,18,15)(28,34,31)(29,35,32)(30,36,33)(37,43,40)(38,44,41)(39,45,42), (10,13,16)(11,14,17)(12,15,18)(19,25,22)(20,26,23)(21,27,24)(37,40,43)(38,41,44)(39,42,45)(46,52,49)(47,53,50)(48,54,51), (1,20,14)(2,27,18)(3,25,13)(4,23,17)(5,21,12)(6,19,16)(7,26,11)(8,24,15)(9,22,10)(28,52,40)(29,50,44)(30,48,39)(31,46,43)(32,53,38)(33,51,42)(34,49,37)(35,47,41)(36,54,45)>;
G:=Group( (1,35)(2,36)(3,28)(4,29)(5,30)(6,31)(7,32)(8,33)(9,34)(10,37)(11,38)(12,39)(13,40)(14,41)(15,42)(16,43)(17,44)(18,45)(19,46)(20,47)(21,48)(22,49)(23,50)(24,51)(25,52)(26,53)(27,54), (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), (1,7,4)(2,8,5)(3,9,6)(10,16,13)(11,17,14)(12,18,15)(28,34,31)(29,35,32)(30,36,33)(37,43,40)(38,44,41)(39,45,42), (10,13,16)(11,14,17)(12,15,18)(19,25,22)(20,26,23)(21,27,24)(37,40,43)(38,41,44)(39,42,45)(46,52,49)(47,53,50)(48,54,51), (1,20,14)(2,27,18)(3,25,13)(4,23,17)(5,21,12)(6,19,16)(7,26,11)(8,24,15)(9,22,10)(28,52,40)(29,50,44)(30,48,39)(31,46,43)(32,53,38)(33,51,42)(34,49,37)(35,47,41)(36,54,45) );
G=PermutationGroup([[(1,35),(2,36),(3,28),(4,29),(5,30),(6,31),(7,32),(8,33),(9,34),(10,37),(11,38),(12,39),(13,40),(14,41),(15,42),(16,43),(17,44),(18,45),(19,46),(20,47),(21,48),(22,49),(23,50),(24,51),(25,52),(26,53),(27,54)], [(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)], [(1,7,4),(2,8,5),(3,9,6),(10,16,13),(11,17,14),(12,18,15),(28,34,31),(29,35,32),(30,36,33),(37,43,40),(38,44,41),(39,45,42)], [(10,13,16),(11,14,17),(12,15,18),(19,25,22),(20,26,23),(21,27,24),(37,40,43),(38,41,44),(39,42,45),(46,52,49),(47,53,50),(48,54,51)], [(1,20,14),(2,27,18),(3,25,13),(4,23,17),(5,21,12),(6,19,16),(7,26,11),(8,24,15),(9,22,10),(28,52,40),(29,50,44),(30,48,39),(31,46,43),(32,53,38),(33,51,42),(34,49,37),(35,47,41),(36,54,45)]])
102 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | ··· | 3J | 3K | ··· | 3P | 6A | 6B | 6C | ··· | 6J | 6K | ··· | 6P | 9A | ··· | 9F | 9G | ··· | 9V | 9W | ··· | 9AH | 18A | ··· | 18F | 18G | ··· | 18V | 18W | ··· | 18AH |
| 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 | 9 | ··· | 9 | 1 | 1 | 3 | ··· | 3 | 9 | ··· | 9 | 1 | ··· | 1 | 3 | ··· | 3 | 9 | ··· | 9 | 1 | ··· | 1 | 3 | ··· | 3 | 9 | ··· | 9 |
102 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 |
| type | + | + | ||||||||||||||||
| image | C1 | C2 | C3 | C3 | C3 | C3 | C3 | C3 | C6 | C6 | C6 | C6 | C6 | C6 | He3 | C2xHe3 | C9.He3 | C2xC9.He3 |
| kernel | C2xC9.He3 | C9.He3 | C2xC3wrC3 | C2xHe3.C3 | C2xHe3:C3 | C2xC3.He3 | C32xC18 | C2xC9oHe3 | C3wrC3 | He3.C3 | He3:C3 | C3.He3 | C32xC9 | C9oHe3 | C18 | C9 | C2 | C1 |
| # reps | 1 | 1 | 6 | 6 | 2 | 4 | 2 | 6 | 6 | 6 | 2 | 4 | 2 | 6 | 6 | 6 | 18 | 18 |
Matrix representation of C2xC9.He3 ►in GL4(F19) generated by
| 18 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 7 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 6 | 9 |
| 0 | 0 | 0 | 9 |
| 7 | 0 | 0 | 0 |
| 0 | 11 | 0 | 0 |
| 0 | 0 | 11 | 8 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 7 | 12 |
| 0 | 0 | 0 | 11 |
| 7 | 0 | 0 | 0 |
| 0 | 0 | 1 | 16 |
| 0 | 11 | 0 | 2 |
| 0 | 10 | 0 | 0 |
G:=sub<GL(4,GF(19))| [18,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[7,0,0,0,0,4,0,0,0,0,6,0,0,0,9,9],[7,0,0,0,0,11,0,0,0,0,11,0,0,0,8,1],[1,0,0,0,0,1,0,0,0,0,7,0,0,0,12,11],[7,0,0,0,0,0,11,10,0,1,0,0,0,16,2,0] >;
C2xC9.He3 in GAP, Magma, Sage, TeX
C_2\times C_9.{\rm He}_3 % in TeX
G:=Group("C2xC9.He3"); // GroupNames label
G:=SmallGroup(486,214);
// by ID
G=gap.SmallGroup(486,214);
# by ID
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,548,237,3250]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^9=c^3=d^3=e^3=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^4,c*d=d*c,e*c*e^-1=b^3*c*d^-1,e*d*e^-1=b^6*d>;
// generators/relations