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## G = C2×C9⋊C18order 324 = 22·34

### Direct product of C2 and C9⋊C18

Series: Derived Chief Lower central Upper central

 Derived series C1 — C9 — C2×C9⋊C18
 Chief series C1 — C3 — C9 — C3×C9 — C9⋊C9 — C9⋊C18 — C2×C9⋊C18
 Lower central C9 — C2×C9⋊C18
 Upper central C1 — C6

Generators and relations for C2×C9⋊C18
G = < a,b,c | a2=b9=c18=1, ab=ba, ac=ca, cbc-1=b2 >

Smallest permutation representation of C2×C9⋊C18
On 36 points
Generators in S36
(1 25)(2 26)(3 27)(4 28)(5 29)(6 30)(7 31)(8 32)(9 33)(10 34)(11 35)(12 36)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)
(1 17 3 13 11 15 7 5 9)(2 4 12 8 10 18 14 16 6)(19 35 21 31 29 33 25 23 27)(20 22 30 26 28 36 32 34 24)
(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)

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

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

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

60 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 3E 6A 6B 6C 6D 6E 6F 6G 6H 6I 9A ··· 9F 9G ··· 9O 18A ··· 18F 18G ··· 18O 18P ··· 18AA order 1 2 2 2 3 3 3 3 3 6 6 6 6 6 6 6 6 6 9 ··· 9 9 ··· 9 18 ··· 18 18 ··· 18 18 ··· 18 size 1 1 9 9 1 1 2 2 2 1 1 2 2 2 9 9 9 9 3 ··· 3 6 ··· 6 3 ··· 3 6 ··· 6 9 ··· 9

60 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 6 6 6 6 type + + + + + + + image C1 C2 C2 C3 C6 C6 C9 C18 C18 S3 D6 C3×S3 S3×C6 S3×C9 S3×C18 C9⋊C6 C2×C9⋊C6 C9⋊C18 C2×C9⋊C18 kernel C2×C9⋊C18 C9⋊C18 C2×C9⋊C9 C6×D9 C3×D9 C3×C18 D18 D9 C18 C3×C18 C3×C9 C3×C6 C32 C6 C3 C6 C3 C2 C1 # reps 1 2 1 2 4 2 6 12 6 1 1 2 2 6 6 1 1 2 2

Matrix representation of C2×C9⋊C18 in GL8(𝔽19)

 18 0 0 0 0 0 0 0 0 18 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 11 0 0 0 0 0 0 0 11 7 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 7 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 11 0 0 0 0 0 0 0 0 11 0
,
 8 4 0 0 0 0 0 0 0 11 0 0 0 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 11 0 0 0 0 0 11 0 0 0 0 0 7 0 0 0 0 0 0 0 0 11 0 0 0 0 0 11 0 0 0 0 0

G:=sub<GL(8,GF(19))| [18,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[11,11,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,7,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,11,0,0,0,0,0,1,0,0],[8,0,0,0,0,0,0,0,4,11,0,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,7,0,0,0,0,0,0,0,0,11,0,0,0,0,0,11,0,0,0,0,0,7,0,0,0,0,0,0,0,0,11,0,0,0,0] >;

C2×C9⋊C18 in GAP, Magma, Sage, TeX

C_2\times C_9\rtimes C_{18}
% in TeX

G:=Group("C2xC9:C18");
// GroupNames label

G:=SmallGroup(324,64);
// by ID

G=gap.SmallGroup(324,64);
# by ID

G:=PCGroup([6,-2,-2,-3,-3,-3,-3,68,5404,1096,208,7781]);
// Polycyclic

G:=Group<a,b,c|a^2=b^9=c^18=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^2>;
// generators/relations

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