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## G = C9×C3⋊S3order 162 = 2·34

### Direct product of C9 and C3⋊S3

Aliases: C9×C3⋊S3, C324C18, C33.6C6, C3⋊(S3×C9), (C3×C9)⋊9S3, (C32×C9)⋊2C2, C32.17(C3×S3), C3.5(C3×C3⋊S3), (C3×C3⋊S3).2C3, SmallGroup(162,39)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C9×C3⋊S3
 Chief series C1 — C3 — C32 — C33 — C32×C9 — C9×C3⋊S3
 Lower central C32 — C9×C3⋊S3
 Upper central C1 — C9

Generators and relations for C9×C3⋊S3
G = < a,b,c,d | a9=b3=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b-1, dcd=c-1 >

Subgroups: 116 in 52 conjugacy classes, 21 normal (9 characteristic)
C1, C2, C3, C3, C3, S3, C6, C9, C9, C32, C32, C32, C18, C3×S3, C3⋊S3, C3×C9, C3×C9, C33, S3×C9, C3×C3⋊S3, C32×C9, C9×C3⋊S3
Quotients: C1, C2, C3, S3, C6, C9, C18, C3×S3, C3⋊S3, S3×C9, C3×C3⋊S3, C9×C3⋊S3

Smallest permutation representation of C9×C3⋊S3
On 54 points
Generators in S54
(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)
(1 28 44)(2 29 45)(3 30 37)(4 31 38)(5 32 39)(6 33 40)(7 34 41)(8 35 42)(9 36 43)(10 20 53)(11 21 54)(12 22 46)(13 23 47)(14 24 48)(15 25 49)(16 26 50)(17 27 51)(18 19 52)
(1 31 41)(2 32 42)(3 33 43)(4 34 44)(5 35 45)(6 36 37)(7 28 38)(8 29 39)(9 30 40)(10 26 47)(11 27 48)(12 19 49)(13 20 50)(14 21 51)(15 22 52)(16 23 53)(17 24 54)(18 25 46)
(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 25)(8 26)(9 27)(10 29)(11 30)(12 31)(13 32)(14 33)(15 34)(16 35)(17 36)(18 28)(37 54)(38 46)(39 47)(40 48)(41 49)(42 50)(43 51)(44 52)(45 53)

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

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

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

C9×C3⋊S3 is a maximal subgroup of
C323Dic9  S32×C9  C325D18  C9⋊S3⋊C9  C32⋊C54  (C3×C9)⋊D9  (C3×C9)⋊3D9  C9⋊He3⋊C2  C33⋊C18  C9⋊(S3×C9)  C924S3  (C32×C9)⋊8S3  C9⋊C92S3  He3.C3⋊S3  He3⋊C32S3  C9○He34S3
C9×C3⋊S3 is a maximal quotient of
C33⋊C18  C9⋊(S3×C9)  C923S3  He3.5C18

54 conjugacy classes

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

54 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 type + + + image C1 C2 C3 C6 C9 C18 S3 C3×S3 S3×C9 kernel C9×C3⋊S3 C32×C9 C3×C3⋊S3 C33 C3⋊S3 C32 C3×C9 C32 C3 # reps 1 1 2 2 6 6 4 8 24

Matrix representation of C9×C3⋊S3 in GL4(𝔽19) generated by

 16 0 0 0 0 16 0 0 0 0 9 0 0 0 0 9
,
 7 0 0 0 0 11 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 7 0 0 0 0 11
,
 0 1 0 0 1 0 0 0 0 0 0 1 0 0 1 0
G:=sub<GL(4,GF(19))| [16,0,0,0,0,16,0,0,0,0,9,0,0,0,0,9],[7,0,0,0,0,11,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,7,0,0,0,0,11],[0,1,0,0,1,0,0,0,0,0,0,1,0,0,1,0] >;

C9×C3⋊S3 in GAP, Magma, Sage, TeX

C_9\times C_3\rtimes S_3
% in TeX

G:=Group("C9xC3:S3");
// GroupNames label

G:=SmallGroup(162,39);
// by ID

G=gap.SmallGroup(162,39);
# by ID

G:=PCGroup([5,-2,-3,-3,-3,-3,36,723,2704]);
// Polycyclic

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

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