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G = C3×C9⋊S3order 162 = 2·34

Direct product of C3 and C9⋊S3

direct product, metabelian, supersoluble, monomial, A-group

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

C1C3×C9 — C3×C9⋊S3
C1C3C32C3×C9C32×C9 — C3×C9⋊S3
C3×C9 — C3×C9⋊S3
C1C3

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 [×2], C3 [×3], C3 [×4], S3 [×4], C6, C9 [×3], C9 [×3], C32 [×2], C32 [×3], C32 [×4], D9 [×3], C3×S3 [×4], C3⋊S3, C3×C9, C3×C9 [×3], C3×C9 [×4], C33, C3×D9 [×3], C9⋊S3, C3×C3⋊S3, C32×C9, C3×C9⋊S3
Quotients: C1, C2, C3, S3 [×4], C6, D9 [×3], C3×S3 [×4], C3⋊S3, C3×D9 [×3], C9⋊S3, C3×C3⋊S3, C3×C9⋊S3

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

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

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

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

C3×C9⋊S3 is a maximal subgroup of
C3×S3×D9  C325D18  C9⋊S3⋊C9  C32⋊D27  (C3×C9)⋊C18  C9⋊S33C9  C3.2(C9⋊D9)  C322D27  (C3×C9)⋊5D9  (C3×C9)⋊6D9  C33.D9  He32D9  3- 1+2⋊D9  C9⋊(S3×C9)  C923S3  C923C6  He33D9  C929C6  C33.5D9  C336D9  He34D9  He3.(C3⋊S3)  C3⋊(He3⋊S3)  (C32×C9).S3  C3≀C3⋊S3  C9○He33S3
C3×C9⋊S3 is a maximal quotient of
C33⋊D9  C924S3  C923C6  He33D9  C929C6  C33.5D9  He3.5D9

45 conjugacy classes

class 1  2 3A3B3C···3N6A6B9A···9AA
order12333···3669···9
size127112···227272···2

45 irreducible representations

dim1111222222
type+++++
imageC1C2C3C6S3S3C3×S3D9C3×S3C3×D9
kernelC3×C9⋊S3C32×C9C9⋊S3C3×C9C3×C9C33C9C32C32C3
# reps11223169218

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

11000
01100
00110
00011
,
5000
0400
00110
0007
,
1000
0100
00110
0007
,
0100
1000
0001
0010
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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