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## G = C2×S3×D9order 216 = 23·33

### Direct product of C2, S3 and D9

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C9 — C2×S3×D9
 Chief series C1 — C3 — C32 — C3×C9 — S3×C9 — S3×D9 — C2×S3×D9
 Lower central C3×C9 — C2×S3×D9
 Upper central C1 — C2

Generators and relations for C2×S3×D9
G = < a,b,c,d,e | a2=b3=c2=d9=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 622 in 106 conjugacy classes, 35 normal (19 characteristic)
C1, C2, C2, C3, C3, C22, S3, S3, C6, C6, C23, C9, C9, C32, D6, D6, C2×C6, D9, D9, C18, C18, C3×S3, C3×S3, C3⋊S3, C3×C6, C22×S3, C3×C9, D18, D18, C2×C18, S32, S3×C6, S3×C6, C2×C3⋊S3, C3×D9, S3×C9, C9⋊S3, C3×C18, C22×D9, C2×S32, S3×D9, C6×D9, S3×C18, C2×C9⋊S3, C2×S3×D9
Quotients: C1, C2, C22, S3, C23, D6, D9, C22×S3, D18, S32, C22×D9, C2×S32, S3×D9, C2×S3×D9

Smallest permutation representation of C2×S3×D9
On 36 points
Generators in S36
(1 14)(2 15)(3 16)(4 17)(5 18)(6 10)(7 11)(8 12)(9 13)(19 28)(20 29)(21 30)(22 31)(23 32)(24 33)(25 34)(26 35)(27 36)
(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 22)(2 23)(3 24)(4 25)(5 26)(6 27)(7 19)(8 20)(9 21)(10 36)(11 28)(12 29)(13 30)(14 31)(15 32)(16 33)(17 34)(18 35)
(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 21)(2 20)(3 19)(4 27)(5 26)(6 25)(7 24)(8 23)(9 22)(10 34)(11 33)(12 32)(13 31)(14 30)(15 29)(16 28)(17 36)(18 35)

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

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

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

C2×S3×D9 is a maximal subgroup of
C36⋊D6  D18⋊D6
C2×S3×D9 is a maximal quotient of
D18.D6  Dic65D9  Dic18⋊S3  D125D9  D12⋊D9  D6.D18  D365S3  Dic9.D6  C36⋊D6  D18.3D6  Dic3.D18  D18.4D6  D18⋊D6

36 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 6A 6B 6C 6D 6E 6F 6G 9A 9B 9C 9D 9E 9F 18A 18B 18C 18D 18E 18F 18G ··· 18L order 1 2 2 2 2 2 2 2 3 3 3 6 6 6 6 6 6 6 9 9 9 9 9 9 18 18 18 18 18 18 18 ··· 18 size 1 1 3 3 9 9 27 27 2 2 4 2 2 4 6 6 18 18 2 2 2 4 4 4 2 2 2 4 4 4 6 ··· 6

36 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 S3 S3 D6 D6 D6 D6 D9 D18 D18 S32 C2×S32 S3×D9 C2×S3×D9 kernel C2×S3×D9 S3×D9 C6×D9 S3×C18 C2×C9⋊S3 D18 S3×C6 D9 C18 C3×S3 C3×C6 D6 S3 C6 C6 C3 C2 C1 # reps 1 4 1 1 1 1 1 2 1 2 1 3 6 3 1 1 3 3

Matrix representation of C2×S3×D9 in GL4(𝔽19) generated by

 18 0 0 0 0 18 0 0 0 0 1 0 0 0 0 1
,
 18 1 0 0 18 0 0 0 0 0 1 0 0 0 0 1
,
 0 1 0 0 1 0 0 0 0 0 18 0 0 0 0 18
,
 1 0 0 0 0 1 0 0 0 0 2 5 0 0 14 7
,
 1 0 0 0 0 1 0 0 0 0 5 12 0 0 17 14
G:=sub<GL(4,GF(19))| [18,0,0,0,0,18,0,0,0,0,1,0,0,0,0,1],[18,18,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,18,0,0,0,0,18],[1,0,0,0,0,1,0,0,0,0,2,14,0,0,5,7],[1,0,0,0,0,1,0,0,0,0,5,17,0,0,12,14] >;

C2×S3×D9 in GAP, Magma, Sage, TeX

C_2\times S_3\times D_9
% in TeX

G:=Group("C2xS3xD9");
// GroupNames label

G:=SmallGroup(216,101);
// by ID

G=gap.SmallGroup(216,101);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-3,-3,1065,453,1444,2603]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^3=c^2=d^9=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations

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