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## G = C2×S3×C3⋊S3order 216 = 23·33

### Direct product of C2, S3 and C3⋊S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C33 — C2×S3×C3⋊S3
 Chief series C1 — C3 — C32 — C33 — S3×C32 — S3×C3⋊S3 — C2×S3×C3⋊S3
 Lower central C33 — C2×S3×C3⋊S3
 Upper central C1 — C2

Generators and relations for C2×S3×C3⋊S3
G = < a,b,c,d,e,f | a2=b3=c2=d3=e3=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, cbc=b-1, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, fdf=d-1, fef=e-1 >

Subgroups: 1252 in 232 conjugacy classes, 56 normal (14 characteristic)
C1, C2, C2, C3, C3, C3, C22, S3, S3, C6, C6, C6, C23, C32, C32, C32, D6, D6, C2×C6, C3×S3, C3×S3, C3⋊S3, C3⋊S3, C3×C6, C3×C6, C3×C6, C22×S3, C33, S32, S3×C6, S3×C6, C2×C3⋊S3, C2×C3⋊S3, C62, S3×C32, C3×C3⋊S3, C33⋊C2, C32×C6, C2×S32, C22×C3⋊S3, S3×C3⋊S3, S3×C3×C6, C6×C3⋊S3, C2×C33⋊C2, C2×S3×C3⋊S3
Quotients: C1, C2, C22, S3, C23, D6, C3⋊S3, C22×S3, S32, C2×C3⋊S3, C2×S32, C22×C3⋊S3, S3×C3⋊S3, C2×S3×C3⋊S3

Smallest permutation representation of C2×S3×C3⋊S3
On 36 points
Generators in S36
(1 10)(2 11)(3 12)(4 13)(5 14)(6 15)(7 16)(8 17)(9 18)(19 28)(20 29)(21 30)(22 31)(23 32)(24 33)(25 34)(26 35)(27 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 28)(2 30)(3 29)(4 34)(5 36)(6 35)(7 31)(8 33)(9 32)(10 19)(11 21)(12 20)(13 25)(14 27)(15 26)(16 22)(17 24)(18 23)
(1 7 4)(2 8 5)(3 9 6)(10 16 13)(11 17 14)(12 18 15)(19 22 25)(20 23 26)(21 24 27)(28 31 34)(29 32 35)(30 33 36)
(1 9 5)(2 7 6)(3 8 4)(10 18 14)(11 16 15)(12 17 13)(19 23 27)(20 24 25)(21 22 26)(28 32 36)(29 33 34)(30 31 35)
(1 28)(2 29)(3 30)(4 31)(5 32)(6 33)(7 34)(8 35)(9 36)(10 19)(11 20)(12 21)(13 22)(14 23)(15 24)(16 25)(17 26)(18 27)

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

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

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

C2×S3×C3⋊S3 is a maximal subgroup of
D6⋊(C32⋊C4)  D64S32  (S3×C6)⋊D6  C3⋊S34D12  C12⋊S32  C6223D6  C2×S33
C2×S3×C3⋊S3 is a maximal quotient of
(C3×D12)⋊S3  D12⋊(C3⋊S3)  C12.39S32  C12.40S32  C329(S3×Q8)  C12.73S32  C12.57S32  C12.58S32  C12⋊S32  C62.90D6  C62.91D6  C62.93D6  C6223D6

36 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A ··· 3E 3F 3G 3H 3I 6A ··· 6E 6F 6G 6H 6I 6J ··· 6Q 6R 6S order 1 2 2 2 2 2 2 2 3 ··· 3 3 3 3 3 6 ··· 6 6 6 6 6 6 ··· 6 6 6 size 1 1 3 3 9 9 27 27 2 ··· 2 4 4 4 4 2 ··· 2 4 4 4 4 6 ··· 6 18 18

36 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 4 4 type + + + + + + + + + + + + image C1 C2 C2 C2 C2 S3 S3 D6 D6 D6 S32 C2×S32 kernel C2×S3×C3⋊S3 S3×C3⋊S3 S3×C3×C6 C6×C3⋊S3 C2×C33⋊C2 S3×C6 C2×C3⋊S3 C3×S3 C3⋊S3 C3×C6 C6 C3 # reps 1 4 1 1 1 4 1 8 2 5 4 4

Matrix representation of C2×S3×C3⋊S3 in GL6(ℤ)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 -1 0 0 0 0 0 0 -1 0 0 0 0 0 0 -1 0 0 0 0 0 0 -1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 -1 -1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 -1 0 0 0 0 0 0 -1 0 0 0 0 0 0 1 0 0 0 0 0 -1 -1
,
 0 -1 0 0 0 0 1 -1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 -1 0 0 0 0 1 -1 0 0 0 0 0 0 -1 1 0 0 0 0 -1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 -1 0 0 0 0 -1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1

G:=sub<GL(6,Integers())| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,0,0,0,0,1,-1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,-1,0,0,0,0,0,-1],[0,1,0,0,0,0,-1,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,-1,-1,0,0,0,0,0,0,-1,-1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,-1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

C2×S3×C3⋊S3 in GAP, Magma, Sage, TeX

C_2\times S_3\times C_3\rtimes S_3
% in TeX

G:=Group("C2xS3xC3:S3");
// GroupNames label

G:=SmallGroup(216,171);
// by ID

G=gap.SmallGroup(216,171);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-3,-3,201,730,5189]);
// Polycyclic

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

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