Finite Group DatabasebetaFinite Group Databasebeta| order | 120 |
| isoclinism | isoclinic class of D3×D5 |
| automorphism | Aut(G) = C22×D3×AGL(1,𝔽5) (order 480) · Inn(G) = D3×D5 · Out(G) = C23 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| derived, Hall | C3×C5 ⋊ C2×C4 |
| Sylow | C5 ⋊ C4×D3 |
| Sylow | C3 ⋊ C4×D5 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D3×D5 |
| commutator, hypocenter | C3×C5 | C2×C4 |
| Fitting, socle, cosocle | C2×C3×C5 | C22 |
| prime radical | C2×D15 | C2 |
| LCS | G ▷ C3×C5 | |
| UCS | C1 ◁ C2 | |
32 conjugacy classes of subgroups.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 60 |
A minimal faithful permutation representation — 2 generators on 12 points:
(1,2,4,5,3)(7,8)(9,11,12,10);(2,3)(4,5)(6,7,8)(9,10,12,11)Polycyclic presentation:
2^3,3,5|a^2=c,b^2=c,e^a=e^4,d^b=d^224 conjugacy classes — view the full character table.
| maximal subgroup of | #6738, #6740, #6769, … |
| maximal normal subgroup of | #6738, #6740, #6769, … |
| maximal quotient of | #6648, #6651, Dic3×Dic5, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(120, 10);