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G = S3xC3xC6order 108 = 22·33

Direct product of C3xC6 and S3

direct product, metabelian, supersoluble, monomial, A-group

Aliases: S3xC3xC6, C3:C62, C33:4C22, C6:(C3xC6), (C3xC6):3C6, C32:4(C2xC6), (C32xC6):1C2, SmallGroup(108,42)

Series: Derived Chief Lower central Upper central

C1C3 — S3xC3xC6
C1C3C32C33S3xC32 — S3xC3xC6
C3 — S3xC3xC6
C1C3xC6

Generators and relations for S3xC3xC6
 G = < a,b,c,d | a3=b6=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 128 in 76 conjugacy classes, 42 normal (10 characteristic)
C1, C2, C2, C3, C3, C3, C22, S3, C6, C6, C6, C32, C32, C32, D6, C2xC6, C3xS3, C3xC6, C3xC6, C3xC6, C33, S3xC6, C62, S3xC32, C32xC6, S3xC3xC6
Quotients: C1, C2, C3, C22, S3, C6, C32, D6, C2xC6, C3xS3, C3xC6, S3xC6, C62, S3xC32, S3xC3xC6

Smallest permutation representation of S3xC3xC6
On 36 points
Generators in S36
(1 29 24)(2 30 19)(3 25 20)(4 26 21)(5 27 22)(6 28 23)(7 14 32)(8 15 33)(9 16 34)(10 17 35)(11 18 36)(12 13 31)
(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)
(1 25 22)(2 26 23)(3 27 24)(4 28 19)(5 29 20)(6 30 21)(7 36 16)(8 31 17)(9 32 18)(10 33 13)(11 34 14)(12 35 15)
(1 34)(2 35)(3 36)(4 31)(5 32)(6 33)(7 27)(8 28)(9 29)(10 30)(11 25)(12 26)(13 21)(14 22)(15 23)(16 24)(17 19)(18 20)

G:=sub<Sym(36)| (1,29,24)(2,30,19)(3,25,20)(4,26,21)(5,27,22)(6,28,23)(7,14,32)(8,15,33)(9,16,34)(10,17,35)(11,18,36)(12,13,31), (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), (1,25,22)(2,26,23)(3,27,24)(4,28,19)(5,29,20)(6,30,21)(7,36,16)(8,31,17)(9,32,18)(10,33,13)(11,34,14)(12,35,15), (1,34)(2,35)(3,36)(4,31)(5,32)(6,33)(7,27)(8,28)(9,29)(10,30)(11,25)(12,26)(13,21)(14,22)(15,23)(16,24)(17,19)(18,20)>;

G:=Group( (1,29,24)(2,30,19)(3,25,20)(4,26,21)(5,27,22)(6,28,23)(7,14,32)(8,15,33)(9,16,34)(10,17,35)(11,18,36)(12,13,31), (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), (1,25,22)(2,26,23)(3,27,24)(4,28,19)(5,29,20)(6,30,21)(7,36,16)(8,31,17)(9,32,18)(10,33,13)(11,34,14)(12,35,15), (1,34)(2,35)(3,36)(4,31)(5,32)(6,33)(7,27)(8,28)(9,29)(10,30)(11,25)(12,26)(13,21)(14,22)(15,23)(16,24)(17,19)(18,20) );

G=PermutationGroup([[(1,29,24),(2,30,19),(3,25,20),(4,26,21),(5,27,22),(6,28,23),(7,14,32),(8,15,33),(9,16,34),(10,17,35),(11,18,36),(12,13,31)], [(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)], [(1,25,22),(2,26,23),(3,27,24),(4,28,19),(5,29,20),(6,30,21),(7,36,16),(8,31,17),(9,32,18),(10,33,13),(11,34,14),(12,35,15)], [(1,34),(2,35),(3,36),(4,31),(5,32),(6,33),(7,27),(8,28),(9,29),(10,30),(11,25),(12,26),(13,21),(14,22),(15,23),(16,24),(17,19),(18,20)]])

S3xC3xC6 is a maximal subgroup of   C33:6D4  C33:7D4

54 conjugacy classes

class 1 2A2B2C3A···3H3I···3Q6A···6H6I···6Q6R···6AG
order12223···33···36···66···66···6
size11331···12···21···12···23···3

54 irreducible representations

dim1111112222
type+++++
imageC1C2C2C3C6C6S3D6C3xS3S3xC6
kernelS3xC3xC6S3xC32C32xC6S3xC6C3xS3C3xC6C3xC6C32C6C3
# reps12181681188

Matrix representation of S3xC3xC6 in GL3(F7) generated by

100
040
004
,
500
020
002
,
100
020
004
,
600
001
010
G:=sub<GL(3,GF(7))| [1,0,0,0,4,0,0,0,4],[5,0,0,0,2,0,0,0,2],[1,0,0,0,2,0,0,0,4],[6,0,0,0,0,1,0,1,0] >;

S3xC3xC6 in GAP, Magma, Sage, TeX

S_3\times C_3\times C_6
% in TeX

G:=Group("S3xC3xC6");
// GroupNames label

G:=SmallGroup(108,42);
// by ID

G=gap.SmallGroup(108,42);
# by ID

G:=PCGroup([5,-2,-2,-3,-3,-3,1804]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^6=c^3=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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