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G = C32xD9order 162 = 2·34

Direct product of C32 and D9

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

Aliases: C32xD9, C33.5S3, C9:3(C3xC6), (C3xC9):14C6, (C32xC9):3C2, C3.1(S3xC32), C32.15(C3xS3), SmallGroup(162,32)

Series: Derived Chief Lower central Upper central

C1C9 — C32xD9
C1C3C9C3xC9C32xC9 — C32xD9
C9 — C32xD9
C1C32

Generators and relations for C32xD9
 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, C3xS3, C3xC6, C3xC9, C3xC9, C33, C3xD9, S3xC32, C32xC9, C32xD9
Quotients: C1, C2, C3, S3, C6, C32, D9, C3xS3, C3xC6, C3xD9, S3xC32, C32xD9

Smallest permutation representation of C32xD9
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)]])

C32xD9 is a maximal subgroup of   D9:He3  D9:3- 1+2  (C32xC9):S3

54 conjugacy classes

class 1  2 3A···3H3I···3Q6A···6H9A···9AA
order123···33···36···69···9
size191···12···29···92···2

54 irreducible representations

dim11112222
type++++
imageC1C2C3C6S3D9C3xS3C3xD9
kernelC32xD9C32xC9C3xD9C3xC9C33C32C32C3
# reps118813824

Matrix representation of C32xD9 in GL3(F19) generated by

700
010
001
,
100
070
007
,
100
093
0017
,
1800
0618
01613
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] >;

C32xD9 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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