direct product, metabelian, nilpotent (class 2), monomial, 3-elementary
Aliases: C2×C9○He3, C18.C32, He3.5C6, C6.3C33, 3- 1+2⋊4C6, (C3×C9)⋊12C6, (C3×C18)⋊5C3, C9.2(C3×C6), (C2×He3).2C3, C32.5(C3×C6), (C3×C6).5C32, C3.3(C32×C6), (C2×3- 1+2)⋊3C3, SmallGroup(162,50)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C9○He3
G = < a,b,c,d,e | a2=b9=c3=e3=1, d1=b6, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=b3c, de=ed >
Subgroups: 82 in 66 conjugacy classes, 58 normal (10 characteristic)
C1, C2, C3, C3, C6, C6, C9, C9, C32, C18, C18, C3×C6, C3×C9, He3, 3- 1+2, C3×C18, C2×He3, C2×3- 1+2, C9○He3, C2×C9○He3
Quotients: C1, C2, C3, C6, C32, C3×C6, C33, C32×C6, C9○He3, C2×C9○He3
►On 54 points
Generators in S54
(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)
(10 16 13)(11 17 14)(12 18 15)(19 22 25)(20 23 26)(21 24 27)(37 43 40)(38 44 41)(39 45 42)(46 49 52)(47 50 53)(48 51 54)
(1 7 4)(2 8 5)(3 9 6)(10 16 13)(11 17 14)(12 18 15)(19 25 22)(20 26 23)(21 27 24)(28 34 31)(29 35 32)(30 36 33)(37 43 40)(38 44 41)(39 45 42)(46 52 49)(47 53 50)(48 54 51)
(1 26 11)(2 27 12)(3 19 13)(4 20 14)(5 21 15)(6 22 16)(7 23 17)(8 24 18)(9 25 10)(28 46 40)(29 47 41)(30 48 42)(31 49 43)(32 50 44)(33 51 45)(34 52 37)(35 53 38)(36 54 39)
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), (10,16,13)(11,17,14)(12,18,15)(19,22,25)(20,23,26)(21,24,27)(37,43,40)(38,44,41)(39,45,42)(46,49,52)(47,50,53)(48,51,54), (1,7,4)(2,8,5)(3,9,6)(10,16,13)(11,17,14)(12,18,15)(19,25,22)(20,26,23)(21,27,24)(28,34,31)(29,35,32)(30,36,33)(37,43,40)(38,44,41)(39,45,42)(46,52,49)(47,53,50)(48,54,51), (1,26,11)(2,27,12)(3,19,13)(4,20,14)(5,21,15)(6,22,16)(7,23,17)(8,24,18)(9,25,10)(28,46,40)(29,47,41)(30,48,42)(31,49,43)(32,50,44)(33,51,45)(34,52,37)(35,53,38)(36,54,39)>;
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), (10,16,13)(11,17,14)(12,18,15)(19,22,25)(20,23,26)(21,24,27)(37,43,40)(38,44,41)(39,45,42)(46,49,52)(47,50,53)(48,51,54), (1,7,4)(2,8,5)(3,9,6)(10,16,13)(11,17,14)(12,18,15)(19,25,22)(20,26,23)(21,27,24)(28,34,31)(29,35,32)(30,36,33)(37,43,40)(38,44,41)(39,45,42)(46,52,49)(47,53,50)(48,54,51), (1,26,11)(2,27,12)(3,19,13)(4,20,14)(5,21,15)(6,22,16)(7,23,17)(8,24,18)(9,25,10)(28,46,40)(29,47,41)(30,48,42)(31,49,43)(32,50,44)(33,51,45)(34,52,37)(35,53,38)(36,54,39) );
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)], [(10,16,13),(11,17,14),(12,18,15),(19,22,25),(20,23,26),(21,24,27),(37,43,40),(38,44,41),(39,45,42),(46,49,52),(47,50,53),(48,51,54)], [(1,7,4),(2,8,5),(3,9,6),(10,16,13),(11,17,14),(12,18,15),(19,25,22),(20,26,23),(21,27,24),(28,34,31),(29,35,32),(30,36,33),(37,43,40),(38,44,41),(39,45,42),(46,52,49),(47,53,50),(48,54,51)], [(1,26,11),(2,27,12),(3,19,13),(4,20,14),(5,21,15),(6,22,16),(7,23,17),(8,24,18),(9,25,10),(28,46,40),(29,47,41),(30,48,42),(31,49,43),(32,50,44),(33,51,45),(34,52,37),(35,53,38),(36,54,39)]])
C2×C9○He3 is a maximal subgroup of
He3.4Dic3 He3.5C12
C2×C9○He3 is a maximal quotient of C18×He3 C18×3- 1+2
66 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | ··· | 3J | 6A | 6B | 6C | ··· | 6J | 9A | ··· | 9F | 9G | ··· | 9V | 18A | ··· | 18F | 18G | ··· | 18V |
| 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 | 1 | ··· | 1 | 3 | ··· | 3 | 1 | ··· | 1 | 3 | ··· | 3 |
66 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 |
| type | + | + | ||||||||
| image | C1 | C2 | C3 | C3 | C3 | C6 | C6 | C6 | C9○He3 | C2×C9○He3 |
| kernel | C2×C9○He3 | C9○He3 | C3×C18 | C2×He3 | C2×3- 1+2 | C3×C9 | He3 | 3- 1+2 | C2 | C1 |
| # reps | 1 | 1 | 8 | 2 | 16 | 8 | 2 | 16 | 6 | 6 |
Matrix representation of C2×C9○He3 ►in GL3(𝔽19) generated by
| 18 | 0 | 0 |
| 0 | 18 | 0 |
| 0 | 0 | 18 |
| 4 | 0 | 0 |
| 0 | 4 | 0 |
| 0 | 0 | 4 |
| 1 | 1 | 12 |
| 0 | 11 | 0 |
| 0 | 0 | 7 |
| 11 | 0 | 0 |
| 0 | 11 | 0 |
| 0 | 0 | 11 |
| 1 | 1 | 12 |
| 0 | 0 | 7 |
| 10 | 18 | 18 |
G:=sub<GL(3,GF(19))| [18,0,0,0,18,0,0,0,18],[4,0,0,0,4,0,0,0,4],[1,0,0,1,11,0,12,0,7],[11,0,0,0,11,0,0,0,11],[1,0,10,1,0,18,12,7,18] >;
C2×C9○He3 in GAP, Magma, Sage, TeX
C_2\times C_9\circ {\rm He}_3 % in TeX
G:=Group("C2xC9oHe3"); // GroupNames label
G:=SmallGroup(162,50);
// by ID
G=gap.SmallGroup(162,50);
# by ID
G:=PCGroup([5,-2,-3,-3,-3,-3,457,78]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^9=c^3=e^3=1,d^1=b^6,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e^-1=b^3*c,d*e=e*d>;
// generators/relations