p-group, metacyclic, nilpotent (class 2), monomial
Aliases: C81⋊C3, C27.C9, C9.C27, C32.C27, C27.2C32, C9.6(C3×C9), (C3×C9).5C9, C3.3(C3×C27), (C3×C27).3C3, SmallGroup(243,24)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
| C1 — C3 — C3 — C3 — C3 — C3 — C3 — C3 — C3 — C3 — C3 — C3 — C3 — C3 — C3 — C3 — C3 — C3 — C3 — C9 — C9 — C9 — C9 — C9 — C9 — C27 — C27 — C81⋊C3 |
Generators and relations for C81⋊C3
G = < a,b | a81=b3=1, bab-1=a28 >
Smallest permutation representation of C81⋊C3
►On 81 points
Generators in S81
(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)
(2 56 29)(3 30 57)(5 59 32)(6 33 60)(8 62 35)(9 36 63)(11 65 38)(12 39 66)(14 68 41)(15 42 69)(17 71 44)(18 45 72)(20 74 47)(21 48 75)(23 77 50)(24 51 78)(26 80 53)(27 54 81)
G:=sub<Sym(81)| (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), (2,56,29)(3,30,57)(5,59,32)(6,33,60)(8,62,35)(9,36,63)(11,65,38)(12,39,66)(14,68,41)(15,42,69)(17,71,44)(18,45,72)(20,74,47)(21,48,75)(23,77,50)(24,51,78)(26,80,53)(27,54,81)>;
G:=Group( (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), (2,56,29)(3,30,57)(5,59,32)(6,33,60)(8,62,35)(9,36,63)(11,65,38)(12,39,66)(14,68,41)(15,42,69)(17,71,44)(18,45,72)(20,74,47)(21,48,75)(23,77,50)(24,51,78)(26,80,53)(27,54,81) );
G=PermutationGroup([[(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)], [(2,56,29),(3,30,57),(5,59,32),(6,33,60),(8,62,35),(9,36,63),(11,65,38),(12,39,66),(14,68,41),(15,42,69),(17,71,44),(18,45,72),(20,74,47),(21,48,75),(23,77,50),(24,51,78),(26,80,53),(27,54,81)]])
C81⋊C3 is a maximal subgroup of
C81⋊C6
99 conjugacy classes
| class | 1 | 3A | 3B | 3C | 3D | 9A | ··· | 9F | 9G | 9H | 9I | 9J | 27A | ··· | 27R | 27S | ··· | 27AD | 81A | ··· | 81BB |
| order | 1 | 3 | 3 | 3 | 3 | 9 | ··· | 9 | 9 | 9 | 9 | 9 | 27 | ··· | 27 | 27 | ··· | 27 | 81 | ··· | 81 |
| size | 1 | 1 | 1 | 3 | 3 | 1 | ··· | 1 | 3 | 3 | 3 | 3 | 1 | ··· | 1 | 3 | ··· | 3 | 3 | ··· | 3 |
99 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 |
| type | + | |||||||
| image | C1 | C3 | C3 | C9 | C9 | C27 | C27 | C81⋊C3 |
| kernel | C81⋊C3 | C81 | C3×C27 | C27 | C3×C9 | C9 | C32 | C1 |
| # reps | 1 | 6 | 2 | 12 | 6 | 36 | 18 | 18 |
Matrix representation of C81⋊C3 ►in GL3(𝔽163) generated by
| 0 | 1 | 0 |
| 0 | 0 | 58 |
| 21 | 0 | 0 |
| 1 | 0 | 0 |
| 0 | 58 | 0 |
| 0 | 0 | 104 |
G:=sub<GL(3,GF(163))| [0,0,21,1,0,0,0,58,0],[1,0,0,0,58,0,0,0,104] >;
C81⋊C3 in GAP, Magma, Sage, TeX
C_{81}\rtimes C_3 % in TeX
G:=Group("C81:C3"); // GroupNames label
G:=SmallGroup(243,24);
// by ID
G=gap.SmallGroup(243,24);
# by ID
G:=PCGroup([5,-3,3,-3,-3,-3,45,841,57,78]);
// Polycyclic
G:=Group<a,b|a^81=b^3=1,b*a*b^-1=a^28>;
// generators/relations
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