Finite Group DatabasebetaFinite Group Databasebeta| order | 15 |
| nilpotency class | 1 |
| isoclinism | isoclinic class of C1 |
| automorphism | Aut(G) = C2×C4 (order 8) · Inn(G) = C1 · Out(G) = C2×C4 |
| holomorph | G ⋊ Aut(G) = D3×AGL(1,𝔽5) |
| homology | Abelianization (H1) = G · Schur multiplier (H2) = C1 |
| cyclic | C15 |
| abelian | C3×C5 |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator, Frattini | C1 | G |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
4 conjugacy classes of subgroups.
| minimal degree | μ(G) = 8 |
| transitive degree | μt(G) = 15 |
A minimal faithful permutation representation — 1 generator on 8 points:
(1,2,3)(4,5,6,7,8)15 conjugacy classes — view the full character table.
| center of | C3×C5×D3, C5×H(3,𝔽3), C5×3−1+2, … |
| derived subgroup of | D15, Dic15, ΓL(1,𝔽16), … |
| Frattini subgroup of | C9×C25, D9×C25, C9×D25, … |
| Fitting subgroup of | C5×D3, C3×D5, D15, … |
| socle of | C5×D3, C3×D5, D15, … |
| maximal subgroup of | C5×D3, C3×D5, D15, … |
| maximal normal subgroup of | C5×D3, C3×D5, D15, … |
| maximal quotient of | C2×C3×C5, C5×C9, C32×C5, … |
| outer automorphism group of | D31, #6923 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15, 1);