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## G = C3⋊S3×C3×C6order 324 = 22·34

### Direct product of C3×C6 and C3⋊S3

Aliases: C3⋊S3×C3×C6, C3319D6, C348C22, C325C62, C6⋊(S3×C32), (C33×C6)⋊2C2, (C32×C6)⋊7S3, (C32×C6)⋊8C6, C3312(C2×C6), C3210(S3×C6), C32(S3×C3×C6), (C3×C6)⋊5(C3×S3), (C3×C6)⋊4(C3×C6), SmallGroup(324,173)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C3⋊S3×C3×C6
 Chief series C1 — C3 — C32 — C33 — C34 — C32×C3⋊S3 — C3⋊S3×C3×C6
 Lower central C32 — C3⋊S3×C3×C6
 Upper central C1 — C3×C6

Generators and relations for C3⋊S3×C3×C6
G = < a,b,c,d,e | a3=b6=c3=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece=c-1, ede=d-1 >

Subgroups: 820 in 356 conjugacy classes, 90 normal (10 characteristic)
C1, C2, C2, C3, C3, C22, S3, C6, C6, C32, C32, C32, D6, C2×C6, C3×S3, C3⋊S3, C3×C6, C3×C6, C3×C6, C33, C33, S3×C6, C2×C3⋊S3, C62, S3×C32, C3×C3⋊S3, C32×C6, C32×C6, C34, S3×C3×C6, C6×C3⋊S3, C32×C3⋊S3, C33×C6, C3⋊S3×C3×C6
Quotients: C1, C2, C3, C22, S3, C6, C32, D6, C2×C6, C3×S3, C3⋊S3, C3×C6, S3×C6, C2×C3⋊S3, C62, S3×C32, C3×C3⋊S3, S3×C3×C6, C6×C3⋊S3, C32×C3⋊S3, C3⋊S3×C3×C6

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

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

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

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

108 conjugacy classes

 class 1 2A 2B 2C 3A ··· 3H 3I ··· 3AR 6A ··· 6H 6I ··· 6AR 6AS ··· 6BH order 1 2 2 2 3 ··· 3 3 ··· 3 6 ··· 6 6 ··· 6 6 ··· 6 size 1 1 9 9 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 9 ··· 9

108 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 C3⋊S3×C3×C6 C32×C3⋊S3 C33×C6 C6×C3⋊S3 C3×C3⋊S3 C32×C6 C32×C6 C33 C3×C6 C32 # reps 1 2 1 8 16 8 4 4 32 32

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

 4 0 0 0 0 4 0 0 0 0 2 0 0 0 0 2
,
 6 0 0 0 0 6 0 0 0 0 3 0 0 0 0 3
,
 4 0 0 0 0 2 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 2 0 0 0 0 4
,
 0 1 0 0 1 0 0 0 0 0 0 1 0 0 1 0
G:=sub<GL(4,GF(7))| [4,0,0,0,0,4,0,0,0,0,2,0,0,0,0,2],[6,0,0,0,0,6,0,0,0,0,3,0,0,0,0,3],[4,0,0,0,0,2,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,2,0,0,0,0,4],[0,1,0,0,1,0,0,0,0,0,0,1,0,0,1,0] >;

C3⋊S3×C3×C6 in GAP, Magma, Sage, TeX

C_3\rtimes S_3\times C_3\times C_6
% in TeX

G:=Group("C3:S3xC3xC6");
// GroupNames label

G:=SmallGroup(324,173);
// by ID

G=gap.SmallGroup(324,173);
# by ID

G:=PCGroup([6,-2,-2,-3,-3,-3,-3,2164,7781]);
// Polycyclic

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

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