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## G = C32×D9order 162 = 2·34

### Direct product of C32 and D9

Aliases: C32×D9, C33.5S3, C93(C3×C6), (C3×C9)⋊14C6, (C32×C9)⋊3C2, C3.1(S3×C32), C32.15(C3×S3), SmallGroup(162,32)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C9 — C32×D9
 Chief series C1 — C3 — C9 — C3×C9 — C32×C9 — C32×D9
 Lower central C9 — C32×D9
 Upper central C1 — C32

Generators and relations for C32×D9
G = < a,b,c,d | a3=b3=c9=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 128 in 52 conjugacy classes, 24 normal (8 characteristic)
C1, C2, C3, C3, C3, S3, C6, C9, C9, C32, C32, C32, D9, C3×S3, C3×C6, C3×C9, C3×C9, C33, C3×D9, S3×C32, C32×C9, C32×D9
Quotients: C1, C2, C3, S3, C6, C32, D9, C3×S3, C3×C6, C3×D9, S3×C32, C32×D9

Smallest permutation representation of C32×D9
On 54 points
Generators in S54
(1 17 20)(2 18 21)(3 10 22)(4 11 23)(5 12 24)(6 13 25)(7 14 26)(8 15 27)(9 16 19)(28 43 49)(29 44 50)(30 45 51)(31 37 52)(32 38 53)(33 39 54)(34 40 46)(35 41 47)(36 42 48)
(1 23 14)(2 24 15)(3 25 16)(4 26 17)(5 27 18)(6 19 10)(7 20 11)(8 21 12)(9 22 13)(28 46 37)(29 47 38)(30 48 39)(31 49 40)(32 50 41)(33 51 42)(34 52 43)(35 53 44)(36 54 45)
(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 30)(2 29)(3 28)(4 36)(5 35)(6 34)(7 33)(8 32)(9 31)(10 43)(11 42)(12 41)(13 40)(14 39)(15 38)(16 37)(17 45)(18 44)(19 52)(20 51)(21 50)(22 49)(23 48)(24 47)(25 46)(26 54)(27 53)

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

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

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

C32×D9 is a maximal subgroup of   D9⋊He3  D9⋊3- 1+2  (C32×C9)⋊S3

54 conjugacy classes

 class 1 2 3A ··· 3H 3I ··· 3Q 6A ··· 6H 9A ··· 9AA order 1 2 3 ··· 3 3 ··· 3 6 ··· 6 9 ··· 9 size 1 9 1 ··· 1 2 ··· 2 9 ··· 9 2 ··· 2

54 irreducible representations

 dim 1 1 1 1 2 2 2 2 type + + + + image C1 C2 C3 C6 S3 D9 C3×S3 C3×D9 kernel C32×D9 C32×C9 C3×D9 C3×C9 C33 C32 C32 C3 # reps 1 1 8 8 1 3 8 24

Matrix representation of C32×D9 in GL3(𝔽19) generated by

 7 0 0 0 1 0 0 0 1
,
 1 0 0 0 7 0 0 0 7
,
 1 0 0 0 9 3 0 0 17
,
 18 0 0 0 6 18 0 16 13
G:=sub<GL(3,GF(19))| [7,0,0,0,1,0,0,0,1],[1,0,0,0,7,0,0,0,7],[1,0,0,0,9,0,0,3,17],[18,0,0,0,6,16,0,18,13] >;

C32×D9 in GAP, Magma, Sage, TeX

C_3^2\times D_9
% in TeX

G:=Group("C3^2xD9");
// GroupNames label

G:=SmallGroup(162,32);
// by ID

G=gap.SmallGroup(162,32);
# by ID

G:=PCGroup([5,-2,-3,-3,-3,-3,1803,138,2704]);
// Polycyclic

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

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