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G = C3×C27order 81 = 34

Abelian group of type [3,27]

direct product, p-group, abelian, monomial

Aliases: C3×C27, SmallGroup(81,5)

Series: Derived Chief Lower central Upper central Jennings

C1 — C3×C27
C1C3C9C3×C9 — C3×C27
C1 — C3×C27
C1 — C3×C27
C1C3C3C3C3C3C3C9C9 — C3×C27

Generators and relations for C3×C27
 G = < a,b | a3=b27=1, ab=ba >


Smallest permutation representation of C3×C27
Regular action on 81 points
Generators in S81
(1 31 74)(2 32 75)(3 33 76)(4 34 77)(5 35 78)(6 36 79)(7 37 80)(8 38 81)(9 39 55)(10 40 56)(11 41 57)(12 42 58)(13 43 59)(14 44 60)(15 45 61)(16 46 62)(17 47 63)(18 48 64)(19 49 65)(20 50 66)(21 51 67)(22 52 68)(23 53 69)(24 54 70)(25 28 71)(26 29 72)(27 30 73)
(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)| (1,31,74)(2,32,75)(3,33,76)(4,34,77)(5,35,78)(6,36,79)(7,37,80)(8,38,81)(9,39,55)(10,40,56)(11,41,57)(12,42,58)(13,43,59)(14,44,60)(15,45,61)(16,46,62)(17,47,63)(18,48,64)(19,49,65)(20,50,66)(21,51,67)(22,52,68)(23,53,69)(24,54,70)(25,28,71)(26,29,72)(27,30,73), (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( (1,31,74)(2,32,75)(3,33,76)(4,34,77)(5,35,78)(6,36,79)(7,37,80)(8,38,81)(9,39,55)(10,40,56)(11,41,57)(12,42,58)(13,43,59)(14,44,60)(15,45,61)(16,46,62)(17,47,63)(18,48,64)(19,49,65)(20,50,66)(21,51,67)(22,52,68)(23,53,69)(24,54,70)(25,28,71)(26,29,72)(27,30,73), (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([(1,31,74),(2,32,75),(3,33,76),(4,34,77),(5,35,78),(6,36,79),(7,37,80),(8,38,81),(9,39,55),(10,40,56),(11,41,57),(12,42,58),(13,43,59),(14,44,60),(15,45,61),(16,46,62),(17,47,63),(18,48,64),(19,49,65),(20,50,66),(21,51,67),(22,52,68),(23,53,69),(24,54,70),(25,28,71),(26,29,72),(27,30,73)], [(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)])

C3×C27 is a maximal subgroup of
C27⋊S3  C272C9  C32⋊C27  C9.5He3  C9.6He3  C9⋊C27  C81⋊C3  C27○He3
C3×C27 is a maximal quotient of
C32⋊C27  C9⋊C27  C81⋊C3

81 conjugacy classes

class 1 3A···3H9A···9R27A···27BB
order13···39···927···27
size11···11···11···1

81 irreducible representations

dim111111
type+
imageC1C3C3C9C9C27
kernelC3×C27C27C3×C9C9C32C3
# reps16212654

Matrix representation of C3×C27 in GL2(𝔽109) generated by

450
063
,
70
026
G:=sub<GL(2,GF(109))| [45,0,0,63],[7,0,0,26] >;

C3×C27 in GAP, Magma, Sage, TeX

C_3\times C_{27}
% in TeX

G:=Group("C3xC27");
// GroupNames label

G:=SmallGroup(81,5);
// by ID

G=gap.SmallGroup(81,5);
# by ID

G:=PCGroup([4,-3,3,-3,-3,36,46]);
// Polycyclic

G:=Group<a,b|a^3=b^27=1,a*b=b*a>;
// generators/relations

Export

Subgroup lattice of C3×C27 in TeX

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