direct product, metabelian, nilpotent (class 2), monomial, 3-elementary
Aliases: C6×3- 1+2, C18⋊C32, C6.2C33, C33.4C6, C9⋊2(C3×C6), (C3×C9)⋊11C6, (C3×C18)⋊4C3, C3.2(C32×C6), (C3×C6).8C32, (C32×C6).2C3, C32.11(C3×C6), SmallGroup(162,49)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C6×3- 1+2
G = < a,b,c | a6=b9=c3=1, ab=ba, ac=ca, cbc-1=b4 >
Subgroups: 100 in 76 conjugacy classes, 64 normal (10 characteristic)
C1, C2, C3, C3, C3, C6, C6, C6, C9, C32, C32, C32, C18, C3×C6, C3×C6, C3×C6, C3×C9, 3- 1+2, C33, C3×C18, C2×3- 1+2, C32×C6, C3×3- 1+2, C6×3- 1+2
Quotients: C1, C2, C3, C6, C32, C3×C6, 3- 1+2, C33, C2×3- 1+2, C32×C6, C3×3- 1+2, C6×3- 1+2
►On 54 points
(1 21 36 15 37 48)(2 22 28 16 38 49)(3 23 29 17 39 50)(4 24 30 18 40 51)(5 25 31 10 41 52)(6 26 32 11 42 53)(7 27 33 12 43 54)(8 19 34 13 44 46)(9 20 35 14 45 47)
(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 30 43)(2 28 38)(3 35 42)(4 33 37)(5 31 41)(6 29 45)(7 36 40)(8 34 44)(9 32 39)(10 52 25)(11 50 20)(12 48 24)(13 46 19)(14 53 23)(15 51 27)(16 49 22)(17 47 26)(18 54 21)
G:=sub<Sym(54)| (1,21,36,15,37,48)(2,22,28,16,38,49)(3,23,29,17,39,50)(4,24,30,18,40,51)(5,25,31,10,41,52)(6,26,32,11,42,53)(7,27,33,12,43,54)(8,19,34,13,44,46)(9,20,35,14,45,47), (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), (1,30,43)(2,28,38)(3,35,42)(4,33,37)(5,31,41)(6,29,45)(7,36,40)(8,34,44)(9,32,39)(10,52,25)(11,50,20)(12,48,24)(13,46,19)(14,53,23)(15,51,27)(16,49,22)(17,47,26)(18,54,21)>;
G:=Group( (1,21,36,15,37,48)(2,22,28,16,38,49)(3,23,29,17,39,50)(4,24,30,18,40,51)(5,25,31,10,41,52)(6,26,32,11,42,53)(7,27,33,12,43,54)(8,19,34,13,44,46)(9,20,35,14,45,47), (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), (1,30,43)(2,28,38)(3,35,42)(4,33,37)(5,31,41)(6,29,45)(7,36,40)(8,34,44)(9,32,39)(10,52,25)(11,50,20)(12,48,24)(13,46,19)(14,53,23)(15,51,27)(16,49,22)(17,47,26)(18,54,21) );
G=PermutationGroup([[(1,21,36,15,37,48),(2,22,28,16,38,49),(3,23,29,17,39,50),(4,24,30,18,40,51),(5,25,31,10,41,52),(6,26,32,11,42,53),(7,27,33,12,43,54),(8,19,34,13,44,46),(9,20,35,14,45,47)], [(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)], [(1,30,43),(2,28,38),(3,35,42),(4,33,37),(5,31,41),(6,29,45),(7,36,40),(8,34,44),(9,32,39),(10,52,25),(11,50,20),(12,48,24),(13,46,19),(14,53,23),(15,51,27),(16,49,22),(17,47,26),(18,54,21)]])
C6×3- 1+2 is a maximal subgroup of
C33.Dic3
66 conjugacy classes
| class | 1 | 2 | 3A | ··· | 3H | 3I | ··· | 3N | 6A | ··· | 6H | 6I | ··· | 6N | 9A | ··· | 9R | 18A | ··· | 18R |
| order | 1 | 2 | 3 | ··· | 3 | 3 | ··· | 3 | 6 | ··· | 6 | 6 | ··· | 6 | 9 | ··· | 9 | 18 | ··· | 18 |
| size | 1 | 1 | 1 | ··· | 1 | 3 | ··· | 3 | 1 | ··· | 1 | 3 | ··· | 3 | 3 | ··· | 3 | 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 | 3- 1+2 | C2×3- 1+2 |
| kernel | C6×3- 1+2 | C3×3- 1+2 | C3×C18 | C2×3- 1+2 | C32×C6 | C3×C9 | 3- 1+2 | C33 | C6 | C3 |
| # reps | 1 | 1 | 6 | 18 | 2 | 6 | 18 | 2 | 6 | 6 |
Matrix representation of C6×3- 1+2 ►in GL4(𝔽19) generated by
| 8 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 |
| 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 8 |
| 7 | 0 | 0 | 0 |
| 0 | 1 | 0 | 18 |
| 0 | 0 | 0 | 11 |
| 0 | 13 | 8 | 18 |
| 1 | 0 | 0 | 0 |
| 0 | 11 | 12 | 7 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 7 |
G:=sub<GL(4,GF(19))| [8,0,0,0,0,8,0,0,0,0,8,0,0,0,0,8],[7,0,0,0,0,1,0,13,0,0,0,8,0,18,11,18],[1,0,0,0,0,11,0,0,0,12,1,0,0,7,0,7] >;
C6×3- 1+2 in GAP, Magma, Sage, TeX
C_6\times 3_-^{1+2} % in TeX
G:=Group("C6xES-(3,1)"); // GroupNames label
G:=SmallGroup(162,49);
// by ID
G=gap.SmallGroup(162,49);
# by ID
G:=PCGroup([5,-2,-3,-3,-3,-3,276,457]);
// Polycyclic
G:=Group<a,b,c|a^6=b^9=c^3=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^4>;
// generators/relations