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G = C32×C9order 81 = 34

Abelian group of type [3,3,9]

direct product, p-group, abelian, monomial

Aliases: C32×C9, SmallGroup(81,11)

Series: Derived Chief Lower central Upper central Jennings

C1 — C32×C9
C1C3C32C33 — C32×C9
C1 — C32×C9
C1 — C32×C9
C1C3C3 — C32×C9

Generators and relations for C32×C9
 G = < a,b,c | a3=b3=c9=1, ab=ba, ac=ca, bc=cb >

Subgroups: 50, all normal (4 characteristic)
C1, C3, C3 [×12], C9 [×9], C32 [×13], C3×C9 [×12], C33, C32×C9
Quotients: C1, C3 [×13], C9 [×9], C32 [×13], C3×C9 [×12], C33, C32×C9

Smallest permutation representation of C32×C9
Regular action on 81 points
Generators in S81
(1 21 13)(2 22 14)(3 23 15)(4 24 16)(5 25 17)(6 26 18)(7 27 10)(8 19 11)(9 20 12)(28 40 56)(29 41 57)(30 42 58)(31 43 59)(32 44 60)(33 45 61)(34 37 62)(35 38 63)(36 39 55)(46 65 80)(47 66 81)(48 67 73)(49 68 74)(50 69 75)(51 70 76)(52 71 77)(53 72 78)(54 64 79)
(1 28 46)(2 29 47)(3 30 48)(4 31 49)(5 32 50)(6 33 51)(7 34 52)(8 35 53)(9 36 54)(10 62 77)(11 63 78)(12 55 79)(13 56 80)(14 57 81)(15 58 73)(16 59 74)(17 60 75)(18 61 76)(19 38 72)(20 39 64)(21 40 65)(22 41 66)(23 42 67)(24 43 68)(25 44 69)(26 45 70)(27 37 71)
(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)(55 56 57 58 59 60 61 62 63)(64 65 66 67 68 69 70 71 72)(73 74 75 76 77 78 79 80 81)

G:=sub<Sym(81)| (1,21,13)(2,22,14)(3,23,15)(4,24,16)(5,25,17)(6,26,18)(7,27,10)(8,19,11)(9,20,12)(28,40,56)(29,41,57)(30,42,58)(31,43,59)(32,44,60)(33,45,61)(34,37,62)(35,38,63)(36,39,55)(46,65,80)(47,66,81)(48,67,73)(49,68,74)(50,69,75)(51,70,76)(52,71,77)(53,72,78)(54,64,79), (1,28,46)(2,29,47)(3,30,48)(4,31,49)(5,32,50)(6,33,51)(7,34,52)(8,35,53)(9,36,54)(10,62,77)(11,63,78)(12,55,79)(13,56,80)(14,57,81)(15,58,73)(16,59,74)(17,60,75)(18,61,76)(19,38,72)(20,39,64)(21,40,65)(22,41,66)(23,42,67)(24,43,68)(25,44,69)(26,45,70)(27,37,71), (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)(55,56,57,58,59,60,61,62,63)(64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81)>;

G:=Group( (1,21,13)(2,22,14)(3,23,15)(4,24,16)(5,25,17)(6,26,18)(7,27,10)(8,19,11)(9,20,12)(28,40,56)(29,41,57)(30,42,58)(31,43,59)(32,44,60)(33,45,61)(34,37,62)(35,38,63)(36,39,55)(46,65,80)(47,66,81)(48,67,73)(49,68,74)(50,69,75)(51,70,76)(52,71,77)(53,72,78)(54,64,79), (1,28,46)(2,29,47)(3,30,48)(4,31,49)(5,32,50)(6,33,51)(7,34,52)(8,35,53)(9,36,54)(10,62,77)(11,63,78)(12,55,79)(13,56,80)(14,57,81)(15,58,73)(16,59,74)(17,60,75)(18,61,76)(19,38,72)(20,39,64)(21,40,65)(22,41,66)(23,42,67)(24,43,68)(25,44,69)(26,45,70)(27,37,71), (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)(55,56,57,58,59,60,61,62,63)(64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81) );

G=PermutationGroup([(1,21,13),(2,22,14),(3,23,15),(4,24,16),(5,25,17),(6,26,18),(7,27,10),(8,19,11),(9,20,12),(28,40,56),(29,41,57),(30,42,58),(31,43,59),(32,44,60),(33,45,61),(34,37,62),(35,38,63),(36,39,55),(46,65,80),(47,66,81),(48,67,73),(49,68,74),(50,69,75),(51,70,76),(52,71,77),(53,72,78),(54,64,79)], [(1,28,46),(2,29,47),(3,30,48),(4,31,49),(5,32,50),(6,33,51),(7,34,52),(8,35,53),(9,36,54),(10,62,77),(11,63,78),(12,55,79),(13,56,80),(14,57,81),(15,58,73),(16,59,74),(17,60,75),(18,61,76),(19,38,72),(20,39,64),(21,40,65),(22,41,66),(23,42,67),(24,43,68),(25,44,69),(26,45,70),(27,37,71)], [(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),(55,56,57,58,59,60,61,62,63),(64,65,66,67,68,69,70,71,72),(73,74,75,76,77,78,79,80,81)])

C32×C9 is a maximal subgroup of
C324D9  C3.C92  C32⋊C27  C32.19He3  C32.20He3  C9.4He3  He3⋊C9  3- 1+2⋊C9  C923C3  C9⋊He3  C32.23C33  C9⋊3- 1+2  C33.31C32  C9.He3
C32×C9 is a maximal quotient of
C923C3  C27○He3

81 conjugacy classes

class 1 3A···3Z9A···9BB
order13···39···9
size11···11···1

81 irreducible representations

dim1111
type+
imageC1C3C3C9
kernelC32×C9C3×C9C33C32
# reps124254

Matrix representation of C32×C9 in GL3(𝔽19) generated by

100
070
007
,
700
0110
0011
,
1600
090
0016
G:=sub<GL(3,GF(19))| [1,0,0,0,7,0,0,0,7],[7,0,0,0,11,0,0,0,11],[16,0,0,0,9,0,0,0,16] >;

C32×C9 in GAP, Magma, Sage, TeX

C_3^2\times C_9
% in TeX

G:=Group("C3^2xC9");
// GroupNames label

G:=SmallGroup(81,11);
// by ID

G=gap.SmallGroup(81,11);
# by ID

G:=PCGroup([4,-3,3,3,-3,108]);
// Polycyclic

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

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