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## G = C32×C9order 81 = 34

### Abelian group of type [3,3,9]

Aliases: C32×C9, SmallGroup(81,11)

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C32×C9
 Chief series C1 — C3 — C32 — C33 — C32×C9
 Lower central C1 — C32×C9
 Upper central C1 — C32×C9
 Jennings C1 — C3 — C3 — 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 ··· 3Z 9A ··· 9BB order 1 3 ··· 3 9 ··· 9 size 1 1 ··· 1 1 ··· 1

81 irreducible representations

 dim 1 1 1 1 type + image C1 C3 C3 C9 kernel C32×C9 C3×C9 C33 C32 # reps 1 24 2 54

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

 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
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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