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 | #6872 ⋊ C2 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C9 | (C3⋊D3)⋊C3 |
| commutator, hypocenter | H(3,𝔽3) | C2×C9 |
| Frattini | C3×C9 | C3×D3 |
| Fitting | #6872 | 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:
(1,8,42,15,49,22,32,2,20,6,27,66,14,48,7,12,11,45,5,26,65,3,24,23,33,10,44)(4,59,75,60,70,53,58,38,61,39,52,78,37,68,40,55,30,67,19,50,69,34,56,46,57,28,51,16,35,81,36,80,29,76,17,77,18,72,64,62,47,63,31,54,43,41,25,79,13,73,21,74,9,71);(1,4,16)(2,9,28)(3,13,34)(5,19,41)(6,18,39)(7,21,46)(8,25,50)(10,30,54)(11,17,38)(12,31,55)(14,37,62)(15,36,60)(20,43,67)(22,40,63)(23,29,53)(24,47,68)(26,52,72)(27,35,59)(32,58,76)(33,57,74)(42,64,78)(44,61,77)(45,51,71)(48,70,80)(49,56,73)(65,75,81)(66,69,79)Polycyclic presentation:
2,3^5|b^3=c,c^3=f,d^a=d^2,e^a=e^2f,d^b=de,e^d=ef66 conjugacy classes — view the full character table.
| maximal subgroup of | #92181⋊C2, C2×#92181, C3×#92181, … |
| maximal normal subgroup of | #92181⋊C2, C2×#92181, C3×#92181, … |
| maximal quotient of | #11757304, C2×#92181, #272⋊C27, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(486, 26);