Finite Group DatabasebetaFinite Group Databasebeta| order | 486 |
| isoclinism | isoclinic class of #4174 |
| automorphism | Aut(G) = C9×((C3⋊D3)⋊C3)⋊C2 (order 972) · Inn(G) = (C3⋊D3)⋊C3 · Out(G) = C2×C9 |
| homology | Abelianization (H1) = C2×C9 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Fitting, Sylow | #6873 ⋊ C2 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C9 | (C3⋊D3)⋊C3 |
| commutator, hypocenter | H(3,𝔽3) | C2×C9 |
| Frattini | C3×C9 | C3×D3 |
| Fitting | #6873 | C2 |
| socle | C3 | (C3⋊D3)⋊C9 |
| prime radical | #272 | C9 |
| cosocle | #737 | C2×C3 |
| LCS | G ▷ H(3,𝔽3) | |
| UCS | C1 ◁ C9 | |
34 conjugacy classes of subgroups.
| minimal degree | μ(G) = 81 |
| transitive degree | μt(G) = 81 |
A minimal faithful permutation representation — 2 generators on 81 points:
(1,7,39,12,45,43,56,2,18,5,23,63,34,69,6,10,9,41,17,49,78,3,21,20,29,27,65)(4,57,74,58,44,42,54,35,59,36,22,79,32,80,37,52,8,70,15,71,62,30,53,50,55,51,40,13,31,81,33,66,19,75,14,76,16,46,67,60,68,61,28,24,47,38,48,77,11,72,25,73,26,64);(1,4,13)(2,8,24)(3,11,30)(5,16,36)(6,19,42)(7,22,46)(9,26,51)(10,28,52)(12,33,58)(14,35,27)(15,38,17)(18,40,64)(20,37,61)(21,44,66)(23,48,71)(25,50,43)(29,55,73)(31,57,49)(32,60,34)(39,62,77)(41,59,76)(45,68,80)(47,70,65)(53,72,69)(54,75,56)(63,74,81)(67,79,78)Polycyclic presentation:
2,3^5|b^3=c,c^3=f,d^a=d^2,e^a=e^2f^2,d^b=de,e^d=ef^266 conjugacy classes — view the full character table.
| maximal subgroup of | #92183⋊C2, C2×#92183, C3×#92183, … |
| maximal normal subgroup of | #92183⋊C2, C2×#92183, C3×#92183, … |
| maximal quotient of | #11757306, C2×#92183, #272⋊C27, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(486, 28);