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G = C2xC3wrC3order 162 = 2·34

Direct product of C2 and C3wrC3

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

Aliases: C2xC3wrC3, He3:2C6, C33:5C6, C6.2He3, 3- 1+2:1C6, (C2xHe3):1C3, (C32xC6):1C3, C3.2(C2xHe3), (C3xC6).1C32, C32.1(C3xC6), (C2x3- 1+2):1C3, SmallGroup(162,28)

Series: Derived Chief Lower central Upper central

C1C32 — C2xC3wrC3
C1C3C32C33C3wrC3 — C2xC3wrC3
C1C3C32 — C2xC3wrC3
C1C6C3xC6 — C2xC3wrC3

Generators and relations for C2xC3wrC3
 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 >

Subgroups: 100 in 40 conjugacy classes, 16 normal (12 characteristic)
Quotients: C1, C2, C3, C6, C32, C3xC6, He3, C2xHe3, C3wrC3, C2xC3wrC3
3C3
3C3
3C3
3C3
9C3
3C6
3C6
3C6
3C6
9C6
3C9
3C9
3C32
3C32
3C32
3C32
3C32
3C18
3C18
3C3xC6
3C3xC6
3C3xC6
3C3xC6
3C3xC6

Permutation representations of C2xC3wrC3
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);

C2xC3wrC3 is a maximal subgroup of   He3:C12  C33:C12  C33:Dic3

34 conjugacy classes

class 1  2 3A3B3C···3J3K3L6A6B6C···6J6K6L9A9B9C9D18A18B18C18D
order12333···333666···666999918181818
size11113···399113···39999999999

34 irreducible representations

dim111111113333
type++
imageC1C2C3C3C3C6C6C6He3C2xHe3C3wrC3C2xC3wrC3
kernelC2xC3wrC3C3wrC3C2xHe3C2x3- 1+2C32xC6He33- 1+2C33C6C3C2C1
# reps112422422266

Matrix representation of C2xC3wrC3 in GL3(F7) generated by

600
060
006
,
100
666
310
,
200
020
002
,
226
450
560
,
100
033
040
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] >;

C2xC3wrC3 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

Export

Subgroup lattice of C2xC3wrC3 in TeX

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