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## G = C3×C9⋊S3order 162 = 2·34

### Direct product of C3 and C9⋊S3

Aliases: C3×C9⋊S3, C323D9, C33.6S3, C3⋊(C3×D9), C93(C3×S3), (C3×C9)⋊15C6, (C3×C9)⋊10S3, (C32×C9)⋊4C2, C32.9(C3⋊S3), C32.16(C3×S3), C3.1(C3×C3⋊S3), SmallGroup(162,38)

Series: Derived Chief Lower central Upper central

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

Generators and relations for C3×C9⋊S3
G = < a,b,c,d | a3=b9=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b-1, dcd=c-1 >

Subgroups: 202 in 55 conjugacy classes, 22 normal (10 characteristic)
C1, C2, C3, C3, C3, S3, C6, C9, C9, C32, C32, C32, D9, C3×S3, C3⋊S3, C3×C9, C3×C9, C3×C9, C33, C3×D9, C9⋊S3, C3×C3⋊S3, C32×C9, C3×C9⋊S3
Quotients: C1, C2, C3, S3, C6, D9, C3×S3, C3⋊S3, C3×D9, C9⋊S3, C3×C3⋊S3, C3×C9⋊S3

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

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

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

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

45 conjugacy classes

 class 1 2 3A 3B 3C ··· 3N 6A 6B 9A ··· 9AA order 1 2 3 3 3 ··· 3 6 6 9 ··· 9 size 1 27 1 1 2 ··· 2 27 27 2 ··· 2

45 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 type + + + + + image C1 C2 C3 C6 S3 S3 C3×S3 D9 C3×S3 C3×D9 kernel C3×C9⋊S3 C32×C9 C9⋊S3 C3×C9 C3×C9 C33 C9 C32 C32 C3 # reps 1 1 2 2 3 1 6 9 2 18

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

 11 0 0 0 0 11 0 0 0 0 11 0 0 0 0 11
,
 5 0 0 0 0 4 0 0 0 0 11 0 0 0 0 7
,
 1 0 0 0 0 1 0 0 0 0 11 0 0 0 0 7
,
 0 1 0 0 1 0 0 0 0 0 0 1 0 0 1 0
G:=sub<GL(4,GF(19))| [11,0,0,0,0,11,0,0,0,0,11,0,0,0,0,11],[5,0,0,0,0,4,0,0,0,0,11,0,0,0,0,7],[1,0,0,0,0,1,0,0,0,0,11,0,0,0,0,7],[0,1,0,0,1,0,0,0,0,0,0,1,0,0,1,0] >;

C3×C9⋊S3 in GAP, Magma, Sage, TeX

C_3\times C_9\rtimes S_3
% in TeX

G:=Group("C3xC9:S3");
// GroupNames label

G:=SmallGroup(162,38);
// by ID

G=gap.SmallGroup(162,38);
# by ID

G:=PCGroup([5,-2,-3,-3,-3,-3,992,282,723,2704]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^9=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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