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## G = C2×C3≀C3order 162 = 2·34

### Direct product of C2 and C3≀C3

direct product, metabelian, nilpotent (class 3), monomial, 3-elementary

Aliases: C2×C3≀C3, He32C6, C335C6, C6.2He3, 3- 1+21C6, (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

 Derived series C1 — C32 — C2×C3≀C3
 Chief series C1 — C3 — C32 — C33 — C3≀C3 — C2×C3≀C3
 Lower central C1 — C3 — C32 — C2×C3≀C3
 Upper central C1 — C6 — C3×C6 — C2×C3≀C3

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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