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## G = C3×C27order 81 = 34

### Abelian group of type [3,27]

Aliases: C3×C27, SmallGroup(81,5)

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C3×C27
 Chief series C1 — C3 — C9 — C3×C9 — C3×C27
 Lower central C1 — C3×C27
 Upper central C1 — C3×C27
 Jennings C1 — C3 — C3 — C3 — C3 — C3 — C3 — C9 — C9 — 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 44 69)(2 45 70)(3 46 71)(4 47 72)(5 48 73)(6 49 74)(7 50 75)(8 51 76)(9 52 77)(10 53 78)(11 54 79)(12 28 80)(13 29 81)(14 30 55)(15 31 56)(16 32 57)(17 33 58)(18 34 59)(19 35 60)(20 36 61)(21 37 62)(22 38 63)(23 39 64)(24 40 65)(25 41 66)(26 42 67)(27 43 68)
(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)| (1,44,69)(2,45,70)(3,46,71)(4,47,72)(5,48,73)(6,49,74)(7,50,75)(8,51,76)(9,52,77)(10,53,78)(11,54,79)(12,28,80)(13,29,81)(14,30,55)(15,31,56)(16,32,57)(17,33,58)(18,34,59)(19,35,60)(20,36,61)(21,37,62)(22,38,63)(23,39,64)(24,40,65)(25,41,66)(26,42,67)(27,43,68), (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( (1,44,69)(2,45,70)(3,46,71)(4,47,72)(5,48,73)(6,49,74)(7,50,75)(8,51,76)(9,52,77)(10,53,78)(11,54,79)(12,28,80)(13,29,81)(14,30,55)(15,31,56)(16,32,57)(17,33,58)(18,34,59)(19,35,60)(20,36,61)(21,37,62)(22,38,63)(23,39,64)(24,40,65)(25,41,66)(26,42,67)(27,43,68), (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([[(1,44,69),(2,45,70),(3,46,71),(4,47,72),(5,48,73),(6,49,74),(7,50,75),(8,51,76),(9,52,77),(10,53,78),(11,54,79),(12,28,80),(13,29,81),(14,30,55),(15,31,56),(16,32,57),(17,33,58),(18,34,59),(19,35,60),(20,36,61),(21,37,62),(22,38,63),(23,39,64),(24,40,65),(25,41,66),(26,42,67),(27,43,68)], [(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 ··· 3H 9A ··· 9R 27A ··· 27BB order 1 3 ··· 3 9 ··· 9 27 ··· 27 size 1 1 ··· 1 1 ··· 1 1 ··· 1

81 irreducible representations

 dim 1 1 1 1 1 1 type + image C1 C3 C3 C9 C9 C27 kernel C3×C27 C27 C3×C9 C9 C32 C3 # reps 1 6 2 12 6 54

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

 45 0 0 63
,
 7 0 0 26
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

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