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## G = C2×C32⋊C18order 324 = 22·34

### Direct product of C2 and C32⋊C18

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C2×C32⋊C18
 Chief series C1 — C3 — C32 — C33 — C32⋊C9 — C32⋊C18 — C2×C32⋊C18
 Lower central C32 — C2×C32⋊C18
 Upper central C1 — C6

Generators and relations for C2×C32⋊C18
G = < a,b,c,d | a2=b3=c3=d18=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1c-1, dcd-1=c-1 >

Subgroups: 253 in 67 conjugacy classes, 25 normal (19 characteristic)
C1, C2, C2, C3, C3, C22, S3, C6, C6, C9, C32, C32, D6, C2×C6, C18, C3×S3, C3⋊S3, C3×C6, C3×C6, C3×C9, C3×C9, C33, C2×C18, S3×C6, C2×C3⋊S3, S3×C9, C3×C18, C3×C18, C3×C3⋊S3, C32×C6, C32⋊C9, S3×C18, C6×C3⋊S3, C32⋊C18, C2×C32⋊C9, C2×C32⋊C18
Quotients: C1, C2, C3, C22, S3, C6, C9, D6, C2×C6, C18, C3×S3, C2×C18, S3×C6, S3×C9, C32⋊C6, S3×C18, C2×C32⋊C6, C32⋊C18, C2×C32⋊C18

Smallest permutation representation of C2×C32⋊C18
On 36 points
Generators in S36
(1 27)(2 28)(3 29)(4 30)(5 31)(6 32)(7 33)(8 34)(9 35)(10 36)(11 19)(12 20)(13 21)(14 22)(15 23)(16 24)(17 25)(18 26)
(2 8 14)(3 9 15)(5 17 11)(6 18 12)(19 31 25)(20 32 26)(22 28 34)(23 29 35)
(1 13 7)(2 8 14)(3 15 9)(4 10 16)(5 17 11)(6 12 18)(19 31 25)(20 26 32)(21 33 27)(22 28 34)(23 35 29)(24 30 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)

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

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

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

60 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 3E 3F 3G 3H 6A 6B 6C 6D 6E 6F 6G 6H 6I 6J 6K 6L 9A ··· 9F 9G ··· 9L 18A ··· 18F 18G ··· 18L 18M ··· 18X order 1 2 2 2 3 3 3 3 3 3 3 3 6 6 6 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 6 6 6 1 1 2 2 2 6 6 6 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 C32⋊C6 C2×C32⋊C6 C32⋊C18 C2×C32⋊C18 kernel C2×C32⋊C18 C32⋊C18 C2×C32⋊C9 C6×C3⋊S3 C3×C3⋊S3 C32×C6 C2×C3⋊S3 C3⋊S3 C3×C6 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×C32⋊C18 in GL6(𝔽19)

 18 0 0 0 0 0 0 18 0 0 0 0 0 0 18 0 0 0 0 0 0 18 0 0 0 0 0 0 18 0 0 0 0 0 0 18
,
 1 18 7 0 3 9 0 7 0 0 0 0 0 0 11 0 0 0 0 0 0 1 0 0 0 0 0 0 11 0 0 0 0 0 0 7
,
 11 0 0 11 14 13 0 11 0 0 0 0 0 0 11 0 0 0 0 0 0 7 0 0 0 0 0 0 7 0 0 0 0 0 0 7
,
 4 12 16 16 0 11 0 0 0 0 0 12 0 0 0 18 0 0 0 12 0 0 0 0 0 0 12 0 0 0 9 11 7 1 3 15

G:=sub<GL(6,GF(19))| [18,0,0,0,0,0,0,18,0,0,0,0,0,0,18,0,0,0,0,0,0,18,0,0,0,0,0,0,18,0,0,0,0,0,0,18],[1,0,0,0,0,0,18,7,0,0,0,0,7,0,11,0,0,0,0,0,0,1,0,0,3,0,0,0,11,0,9,0,0,0,0,7],[11,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,11,0,0,7,0,0,14,0,0,0,7,0,13,0,0,0,0,7],[4,0,0,0,0,9,12,0,0,12,0,11,16,0,0,0,12,7,16,0,18,0,0,1,0,0,0,0,0,3,11,12,0,0,0,15] >;

C2×C32⋊C18 in GAP, Magma, Sage, TeX

C_2\times C_3^2\rtimes C_{18}
% in TeX

G:=Group("C2xC3^2:C18");
// GroupNames label

G:=SmallGroup(324,62);
// by ID

G=gap.SmallGroup(324,62);
# by ID

G:=PCGroup([6,-2,-2,-3,-3,-3,-3,68,2164,1096,7781]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^3=c^3=d^18=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1*c^-1,d*c*d^-1=c^-1>;
// generators/relations

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