Finite Group DatabasebetaFinite Group Databasebeta| order | 486 |
| isoclinism | isoclinic class of #4174 |
| automorphism | Aut(G) = C32×((C3⋊D3)⋊C3)⋊C2 (order 972) · Inn(G) = (C3⋊D3)⋊C3 · Out(G) = C2×C32 |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C3 |
No aliases for this group.
| Fitting, Sylow | #6908 ⋊ C2 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C9 | (C3⋊D3)⋊C3 |
| commutator, cosocle, hypocenter | H(3,𝔽3) | C2×C32 |
| Frattini | C32 | C32×D3 |
| Fitting | #6908 | C2 |
| socle | C3 | C3×(C3⋊D3)⋊C3 |
| LCS | G ▷ H(3,𝔽3) | |
| UCS | C1 ◁ C9 | |
77 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 27 |
A minimal faithful permutation representation — 2 generators on 27 points:
(1,9,14,5,8,6,4,2,15)(3,24,23,19,18,21,12,16,22,10,17,27,11,25,13,20,7,26);(1,3,19,4,12,10,5,11,20)(2,7,24,8,18,16,9,17,25)(6,13,26,14,23,21,15,22,27)Polycyclic presentation:
2,3^5|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 | #92280⋊C2, C2×#92280, #13091021, … |
| maximal normal subgroup of | #92280⋊C2, C2×#92280, #13091021, … |
| maximal quotient of | #11757584, C2×#92280, #13090711, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(486, 125);