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## G = C33⋊C18order 486 = 2·35

### 4th semidirect product of C33 and C18 acting via C18/C3=C6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C33 — C33⋊C18
 Chief series C1 — C3 — C32 — C33 — C34 — C3×C32⋊C9 — C33⋊C18
 Lower central C33 — C33⋊C18
 Upper central C1 — C3

Generators and relations for C33⋊C18
G = < a,b,c,d | a3=b3=c3=d18=1, ab=ba, ac=ca, dad-1=a-1c-1, bc=cb, dbd-1=b-1, dcd-1=c-1 >

Subgroups: 732 in 135 conjugacy classes, 29 normal (14 characteristic)
C1, C2, C3, C3, C3, S3, C6, C9, C32, C32, C32, C18, C3×S3, C3⋊S3, C3×C9, C3×C9, C33, C33, C33, S3×C9, C3×C3⋊S3, C33⋊C2, C32⋊C9, C32⋊C9, C32×C9, C32×C9, C34, C32⋊C18, C9×C3⋊S3, C3×C33⋊C2, C3×C32⋊C9, C33⋊C18
Quotients: C1, C2, C3, S3, C6, C9, C18, C3×S3, C3⋊S3, S3×C9, C32⋊C6, C3×C3⋊S3, C32⋊C18, C9×C3⋊S3, He34S3, C33⋊C18

Smallest permutation representation of C33⋊C18
On 54 points
Generators in S54
(2 8 14)(3 9 15)(5 17 11)(6 18 12)(20 32 26)(21 33 27)(23 29 35)(24 30 36)(37 49 43)(39 45 51)(40 46 52)(42 54 48)
(1 34 50)(2 51 35)(3 36 52)(4 53 19)(5 20 54)(6 37 21)(7 22 38)(8 39 23)(9 24 40)(10 41 25)(11 26 42)(12 43 27)(13 28 44)(14 45 29)(15 30 46)(16 47 31)(17 32 48)(18 49 33)
(1 13 7)(2 8 14)(3 15 9)(4 10 16)(5 17 11)(6 12 18)(19 25 31)(20 32 26)(21 27 33)(22 34 28)(23 29 35)(24 36 30)(37 43 49)(38 50 44)(39 45 51)(40 52 46)(41 47 53)(42 54 48)
(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)(37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54)

G:=sub<Sym(54)| (2,8,14)(3,9,15)(5,17,11)(6,18,12)(20,32,26)(21,33,27)(23,29,35)(24,30,36)(37,49,43)(39,45,51)(40,46,52)(42,54,48), (1,34,50)(2,51,35)(3,36,52)(4,53,19)(5,20,54)(6,37,21)(7,22,38)(8,39,23)(9,24,40)(10,41,25)(11,26,42)(12,43,27)(13,28,44)(14,45,29)(15,30,46)(16,47,31)(17,32,48)(18,49,33), (1,13,7)(2,8,14)(3,15,9)(4,10,16)(5,17,11)(6,12,18)(19,25,31)(20,32,26)(21,27,33)(22,34,28)(23,29,35)(24,36,30)(37,43,49)(38,50,44)(39,45,51)(40,52,46)(41,47,53)(42,54,48), (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)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54)>;

G:=Group( (2,8,14)(3,9,15)(5,17,11)(6,18,12)(20,32,26)(21,33,27)(23,29,35)(24,30,36)(37,49,43)(39,45,51)(40,46,52)(42,54,48), (1,34,50)(2,51,35)(3,36,52)(4,53,19)(5,20,54)(6,37,21)(7,22,38)(8,39,23)(9,24,40)(10,41,25)(11,26,42)(12,43,27)(13,28,44)(14,45,29)(15,30,46)(16,47,31)(17,32,48)(18,49,33), (1,13,7)(2,8,14)(3,15,9)(4,10,16)(5,17,11)(6,12,18)(19,25,31)(20,32,26)(21,27,33)(22,34,28)(23,29,35)(24,36,30)(37,43,49)(38,50,44)(39,45,51)(40,52,46)(41,47,53)(42,54,48), (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)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54) );

G=PermutationGroup([[(2,8,14),(3,9,15),(5,17,11),(6,18,12),(20,32,26),(21,33,27),(23,29,35),(24,30,36),(37,49,43),(39,45,51),(40,46,52),(42,54,48)], [(1,34,50),(2,51,35),(3,36,52),(4,53,19),(5,20,54),(6,37,21),(7,22,38),(8,39,23),(9,24,40),(10,41,25),(11,26,42),(12,43,27),(13,28,44),(14,45,29),(15,30,46),(16,47,31),(17,32,48),(18,49,33)], [(1,13,7),(2,8,14),(3,15,9),(4,10,16),(5,17,11),(6,12,18),(19,25,31),(20,32,26),(21,27,33),(22,34,28),(23,29,35),(24,36,30),(37,43,49),(38,50,44),(39,45,51),(40,52,46),(41,47,53),(42,54,48)], [(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),(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54)]])

63 conjugacy classes

 class 1 2 3A 3B 3C ··· 3N 3O ··· 3W 6A 6B 9A ··· 9F 9G ··· 9AD 18A ··· 18F order 1 2 3 3 3 ··· 3 3 ··· 3 6 6 9 ··· 9 9 ··· 9 18 ··· 18 size 1 27 1 1 2 ··· 2 6 ··· 6 27 27 3 ··· 3 6 ··· 6 27 ··· 27

63 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 6 6 type + + + + + image C1 C2 C3 C6 C9 C18 S3 S3 C3×S3 S3×C9 C32⋊C6 C32⋊C18 kernel C33⋊C18 C3×C32⋊C9 C3×C33⋊C2 C34 C33⋊C2 C33 C32⋊C9 C32×C9 C33 C32 C32 C3 # reps 1 1 2 2 6 6 3 1 8 24 3 6

Matrix representation of C33⋊C18 in GL8(𝔽19)

 0 18 0 0 0 0 0 0 1 18 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 11 12 11 0 0 0 0 0 0 0 0 1 0 0 0 0 0 7 0 0 11 0 0 0 18 0 0 0 0 7
,
 18 1 0 0 0 0 0 0 18 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 7 0 0 0 0 0 7 0 0 11 0 0 0 0 0 7 0 0 11 0 0 0 12 12 0 0 0 11
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 7 0 0 0 0 0 7 0 0 11 0 0 0 0 0 7 0 0 11 0 0 0 12 12 0 0 0 11
,
 0 9 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 0 8 8 0 0 0 9 0 0 7 0 0 4 0 0 0 0 12 12 0 7 7 11 0 0 0 0 11 0 0 8 0 0 0 0 0 12 0 0 0 0 12 12 0 7 0 11

G:=sub<GL(8,GF(19))| [0,1,0,0,0,0,0,0,18,18,0,0,0,0,0,0,0,0,1,0,11,0,0,18,0,0,0,7,12,0,7,0,0,0,0,0,11,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,7],[18,18,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,7,0,0,7,0,12,0,0,0,7,0,0,7,12,0,0,0,0,7,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,11],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,7,0,0,7,0,12,0,0,0,7,0,0,7,12,0,0,0,0,7,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,11],[0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,0,8,7,12,0,0,12,0,0,8,0,12,0,0,12,0,0,0,0,0,11,0,0,0,0,0,4,7,0,12,7,0,0,0,0,7,0,0,0,0,0,9,0,11,8,0,11] >;

C33⋊C18 in GAP, Magma, Sage, TeX

C_3^3\rtimes C_{18}
% in TeX

G:=Group("C3^3:C18");
// GroupNames label

G:=SmallGroup(486,136);
// by ID

G=gap.SmallGroup(486,136);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,43,867,2169,3244,11669]);
// Polycyclic

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

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