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## G = C6×C3⋊S3order 108 = 22·33

### Direct product of C6 and C3⋊S3

Aliases: C6×C3⋊S3, C327D6, C335C22, C6⋊(C3×S3), C32(S3×C6), (C3×C6)⋊4C6, (C3×C6)⋊3S3, (C32×C6)⋊2C2, C325(C2×C6), SmallGroup(108,43)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C6×C3⋊S3
 Chief series C1 — C3 — C32 — C33 — C3×C3⋊S3 — C6×C3⋊S3
 Lower central C32 — C6×C3⋊S3
 Upper central C1 — C6

Generators and relations for C6×C3⋊S3
G = < a,b,c,d | a6=b3=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b-1, dcd=c-1 >

Subgroups: 188 in 76 conjugacy classes, 30 normal (10 characteristic)
C1, C2, C2 [×2], C3, C3 [×4], C3 [×4], C22, S3 [×8], C6, C6 [×4], C6 [×6], C32, C32 [×4], C32 [×4], D6 [×4], C2×C6, C3×S3 [×8], C3⋊S3 [×2], C3×C6, C3×C6 [×4], C3×C6 [×4], C33, S3×C6 [×4], C2×C3⋊S3, C3×C3⋊S3 [×2], C32×C6, C6×C3⋊S3
Quotients: C1, C2 [×3], C3, C22, S3 [×4], C6 [×3], D6 [×4], C2×C6, C3×S3 [×4], C3⋊S3, S3×C6 [×4], C2×C3⋊S3, C3×C3⋊S3, C6×C3⋊S3

Smallest permutation representation of C6×C3⋊S3
On 36 points
Generators in S36
(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 17 34)(8 18 35)(9 13 36)(10 14 31)(11 15 32)(12 16 33)
(1 27 20)(2 28 21)(3 29 22)(4 30 23)(5 25 24)(6 26 19)(7 15 36)(8 16 31)(9 17 32)(10 18 33)(11 13 34)(12 14 35)
(1 17)(2 18)(3 13)(4 14)(5 15)(6 16)(7 25)(8 26)(9 27)(10 28)(11 29)(12 30)(19 31)(20 32)(21 33)(22 34)(23 35)(24 36)

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

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

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

C6×C3⋊S3 is a maximal subgroup of   C336D4  C338D4  C339(C2×C4)  C339D4  S32×C6

36 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C ··· 3N 6A 6B 6C ··· 6N 6O 6P 6Q 6R order 1 2 2 2 3 3 3 ··· 3 6 6 6 ··· 6 6 6 6 6 size 1 1 9 9 1 1 2 ··· 2 1 1 2 ··· 2 9 9 9 9

36 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 type + + + + + image C1 C2 C2 C3 C6 C6 S3 D6 C3×S3 S3×C6 kernel C6×C3⋊S3 C3×C3⋊S3 C32×C6 C2×C3⋊S3 C3⋊S3 C3×C6 C3×C6 C32 C6 C3 # reps 1 2 1 2 4 2 4 4 8 8

Matrix representation of C6×C3⋊S3 in GL4(𝔽7) generated by

 6 0 0 0 0 6 0 0 0 0 5 0 0 0 0 5
,
 4 0 0 0 0 2 0 0 0 0 1 0 0 0 0 1
,
 4 0 0 0 0 2 0 0 0 0 4 0 0 0 0 2
,
 0 1 0 0 1 0 0 0 0 0 0 1 0 0 1 0
G:=sub<GL(4,GF(7))| [6,0,0,0,0,6,0,0,0,0,5,0,0,0,0,5],[4,0,0,0,0,2,0,0,0,0,1,0,0,0,0,1],[4,0,0,0,0,2,0,0,0,0,4,0,0,0,0,2],[0,1,0,0,1,0,0,0,0,0,0,1,0,0,1,0] >;

C6×C3⋊S3 in GAP, Magma, Sage, TeX

C_6\times C_3\rtimes S_3
% in TeX

G:=Group("C6xC3:S3");
// GroupNames label

G:=SmallGroup(108,43);
// by ID

G=gap.SmallGroup(108,43);
# by ID

G:=PCGroup([5,-2,-2,-3,-3,-3,483,1804]);
// Polycyclic

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

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