direct product, metabelian, nilpotent (class 3), monomial, 3-elementary
Aliases: C2×C3≀C3, He3⋊2C6, C33⋊5C6, C6.2He3, 3- 1+2⋊1C6, (C2×He3)⋊1C3, (C32×C6)⋊1C3, C3.2(C2×He3), (C3×C6).1C32, C32.1(C3×C6), (C2×3- 1+2)⋊1C3, SmallGroup(162,28)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C3≀C3
G = < a,b,c,d,e | a2=b3=c3=d3=e3=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=bc-1, be=eb, cd=dc, ce=ec, ede-1=bc-1d >
Permutation representations of C2×C3≀C3
►On 18 points - transitive group 18T75
Generators in S18
(1 6)(2 4)(3 5)(7 16)(8 17)(9 18)(10 15)(11 13)(12 14)
(7 8 9)(10 11 12)(13 14 15)(16 17 18)
(1 5 4)(2 6 3)(7 9 8)(10 11 12)(13 14 15)(16 18 17)
(1 16 13)(2 8 10)(3 9 12)(4 17 15)(5 18 14)(6 7 11)
(1 4 5)(2 3 6)(10 12 11)(13 15 14)
G:=sub<Sym(18)| (1,6)(2,4)(3,5)(7,16)(8,17)(9,18)(10,15)(11,13)(12,14), (7,8,9)(10,11,12)(13,14,15)(16,17,18), (1,5,4)(2,6,3)(7,9,8)(10,11,12)(13,14,15)(16,18,17), (1,16,13)(2,8,10)(3,9,12)(4,17,15)(5,18,14)(6,7,11), (1,4,5)(2,3,6)(10,12,11)(13,15,14)>;
G:=Group( (1,6)(2,4)(3,5)(7,16)(8,17)(9,18)(10,15)(11,13)(12,14), (7,8,9)(10,11,12)(13,14,15)(16,17,18), (1,5,4)(2,6,3)(7,9,8)(10,11,12)(13,14,15)(16,18,17), (1,16,13)(2,8,10)(3,9,12)(4,17,15)(5,18,14)(6,7,11), (1,4,5)(2,3,6)(10,12,11)(13,15,14) );
G=PermutationGroup([[(1,6),(2,4),(3,5),(7,16),(8,17),(9,18),(10,15),(11,13),(12,14)], [(7,8,9),(10,11,12),(13,14,15),(16,17,18)], [(1,5,4),(2,6,3),(7,9,8),(10,11,12),(13,14,15),(16,18,17)], [(1,16,13),(2,8,10),(3,9,12),(4,17,15),(5,18,14),(6,7,11)], [(1,4,5),(2,3,6),(10,12,11),(13,15,14)]])
G:=TransitiveGroup(18,75);
C2×C3≀C3 is a maximal subgroup of
He3⋊C12 C33⋊C12 C33⋊Dic3
34 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | ··· | 3J | 3K | 3L | 6A | 6B | 6C | ··· | 6J | 6K | 6L | 9A | 9B | 9C | 9D | 18A | 18B | 18C | 18D |
| order | 1 | 2 | 3 | 3 | 3 | ··· | 3 | 3 | 3 | 6 | 6 | 6 | ··· | 6 | 6 | 6 | 9 | 9 | 9 | 9 | 18 | 18 | 18 | 18 |
| size | 1 | 1 | 1 | 1 | 3 | ··· | 3 | 9 | 9 | 1 | 1 | 3 | ··· | 3 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 |
34 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 |
| type | + | + | ||||||||||
| image | C1 | C2 | C3 | C3 | C3 | C6 | C6 | C6 | He3 | C2×He3 | C3≀C3 | C2×C3≀C3 |
| kernel | C2×C3≀C3 | C3≀C3 | C2×He3 | C2×3- 1+2 | C32×C6 | He3 | 3- 1+2 | C33 | C6 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 4 | 2 | 2 | 4 | 2 | 2 | 2 | 6 | 6 |
Matrix representation of C2×C3≀C3 ►in GL3(𝔽7) generated by
| 6 | 0 | 0 |
| 0 | 6 | 0 |
| 0 | 0 | 6 |
| 1 | 0 | 0 |
| 6 | 6 | 6 |
| 3 | 1 | 0 |
| 2 | 0 | 0 |
| 0 | 2 | 0 |
| 0 | 0 | 2 |
| 2 | 2 | 6 |
| 4 | 5 | 0 |
| 5 | 6 | 0 |
| 1 | 0 | 0 |
| 0 | 3 | 3 |
| 0 | 4 | 0 |
G:=sub<GL(3,GF(7))| [6,0,0,0,6,0,0,0,6],[1,6,3,0,6,1,0,6,0],[2,0,0,0,2,0,0,0,2],[2,4,5,2,5,6,6,0,0],[1,0,0,0,3,4,0,3,0] >;
C2×C3≀C3 in GAP, Magma, Sage, TeX
C_2\times C_3\wr C_3
% in TeX
G:=Group("C2xC3wrC3"); // GroupNames label
G:=SmallGroup(162,28);
// by ID
G=gap.SmallGroup(162,28);
# by ID
G:=PCGroup([5,-2,-3,-3,-3,-3,187,728]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^3=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,d*b*d^-1=b*c^-1,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=b*c^-1*d>;
// generators/relations
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