Finite Group DatabasebetaFinite Group Databasebeta| order | 240 |
| automorphism | Aut(G) = C2×D3×D4×AGL(1,𝔽5) (order 1920) · Inn(G) = C2×D3×D5 · Out(G) = C22×C4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C3×C4×C5 ⋊ C22 |
| Hall | C3×C5 ⋊ C2×D4 |
| Sylow | C5 ⋊ D3×D4 |
| Sylow | C3 ⋊ D4×D5 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×D3×D5 |
| commutator, socle, cosocle | C2×C3×C5 | C23 |
| Fitting | C3×C4×C5 | C22 |
| hypercenter | C4 | D3×D5 |
| hypocenter | C3×C5 | C2×D4 |
| LCS | G ▷ C2×C3×C5 ▷ C3×C5 | |
| UCS | C1 ◁ C2 ◁ C4 | |
108 conjugacy classes of subgroups.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 60 |
A minimal faithful permutation representation — 3 generators on 12 points:
(1,2,4,5,3)(6,7,8)(9,10,11,12);(7,8)(10,12);(2,3)(4,5)(10,12)Polycyclic presentation:
2^4,3,5|c^2=d,c^a=cd,f^a=f^4,c^b=cd,e^b=e^233 conjugacy classes — view the full character table.
| maximal subgroup of | #91255, #91256, #91257, … |
| maximal normal subgroup of | #91255, #91256, #91257, … |
| maximal quotient of | #91255, #91256, #91257, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(240, 138);