Finite Group DatabasebetaFinite Group Databasebeta| order | 18 |
| nilpotency class | 1 |
| isoclinism | isoclinic class of C1 |
| automorphism | Aut(G) = GL(2,𝔽3) (order 48) · Inn(G) = C1 · Out(G) = GL(2,𝔽3) |
| holomorph | G ⋊ Aut(G) = C2×AGL(2,𝔽3) |
| homology | Abelianization (H1) = G · Schur multiplier (H2) = C3 |
| abelian | C2×C32 |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator, Frattini | C1 | G |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
12 conjugacy classes of subgroups.
| minimal degree | μ(G) = 8 |
| transitive degree | μt(G) = 18 |
A minimal faithful permutation representation — 2 generators on 8 points:
(1,2)(3,4,5);(6,7,8)18 conjugacy classes — view the full character table.
| center of | C32×D4, C32×Q8, C32×Dic3, … |
| derived subgroup of | #625, #626, #627, … |
| Frattini subgroup of | C9×Dic9, (C3⋊Dic3)⋊C9, #66316, … |
| Fitting subgroup of | C3×Dic3, C3⋊Dic3, C2×C3×D3, … |
| socle of | C3×Dic3, C3⋊Dic3, C32×C4, … |
| maximal subgroup of | C3×Dic3, C3⋊Dic3, C32×C4, … |
| maximal normal subgroup of | C3×Dic3, C3⋊Dic3, C32×C4, … |
| maximal quotient of | C32×C4, C22×C32, C2×C3×C9, … |
| outer automorphism group of | #733, C7×D7, C7×D9, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18, 5);