Finite Group DatabasebetaFinite Group Databasebeta| order | 30 |
| 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) = C2×D3×AGL(1,𝔽5) |
| homology | Abelianization (H1) = G · Schur multiplier (H2) = C1 |
| cyclic | C30 |
| abelian | C2×C3×C5 |
| multiplicative | (ℤ/31)ˣ, (ℤ/62)ˣ |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator, Frattini | C1 | G |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
8 conjugacy classes of subgroups.
| minimal degree | μ(G) = 10 |
| transitive degree | μt(G) = 30 |
A minimal faithful permutation representation — 1 generator on 10 points:
(1,2)(3,4,5)(6,7,8,9,10)30 conjugacy classes — view the full character table.
| center of | C3×C5×D4, C3×C5×Q8, C3×C5×Dic3, … |
| derived subgroup of | #1201, #1202, #1203, … |
| Frattini subgroup of | C25×Dic9, C9×Dic25, Dic225, … |
| Fitting subgroup of | C5×Dic3, C3×Dic5, Dic15, … |
| socle of | C5×Dic3, C3×Dic5, Dic15, … |
| maximal subgroup of | C5×Dic3, C3×Dic5, Dic15, … |
| maximal normal subgroup of | C5×Dic3, C3×Dic5, Dic15, … |
| maximal quotient of | C3×C4×C5, C22×C3×C5, C2×C5×C9, … |
| automorphism group of | C31, C2×C31 |
| outer automorphism group of | C31, C2×C31, D61, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(30, 4);