direct product, p-group, abelian, monomial
Aliases: C3×C27, SmallGroup(81,5)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
| C1 — C3×C27 |
| C1 — C3×C27 |
| C1 — 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 C27⋊2C9 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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