Finite Group DatabasebetaFinite Group Databasebeta| order | 27 |
| nilpotency class | 1 |
| isoclinism | isoclinic class of C1 |
| automorphism | Aut(G) = C2×SL(3,𝔽3) (order 11232) · Inn(G) = C1 · Out(G) = C2×SL(3,𝔽3) |
| homology | Abelianization (H1) = G · Schur multiplier (H2) = C33 |
| abelian | C33 |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator, Frattini | C1 | G |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
| LECS | G ▷ C1 | |
28 conjugacy classes of subgroups.
| minimal degree | μ(G) = 9 |
| transitive degree | μt(G) = 27 |
A minimal faithful permutation representation — 3 generators on 9 points:
(1,2,3);(4,5,6);(7,8,9)27 conjugacy classes — view the full character table.
| center of | C33×D3, #6855, C3×C32⋊C9, … |
| derived subgroup of | C3⋊(C3⋊D3), C3⋊(C3⋊Dic3), C3⋊#157, … |
| Frattini subgroup of | #6855, #6856, #6857, … |
| Fitting subgroup of | C32×D3, C3×C3⋊D3, C3⋊(C3⋊D3), … |
| socle of | C32×D3, C3×C3⋊D3, C3⋊(C3⋊D3), … |
| maximal subgroup of | C32×D3, C3×C3⋊D3, C3⋊(C3⋊D3), … |
| maximal normal subgroup of | C32×D3, C3×C3⋊D3, C3⋊(C3⋊D3), … |
| maximal quotient of | C2×C33, C32×C9, C3×H(3,𝔽3), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(27, 5);