Finite Group DatabasebetaFinite Group Databasebeta| order | 240 |
| automorphism | Aut(G) = C2×D3×AGL(1,𝔽5)×S4 (order 5760) · Inn(G) = C22×D15 · Out(G) = C2×C4×D3 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C3×C5×Q8 ⋊ C2 |
| Hall | C3×C5 ⋊ #41 |
| Sylow | C5 ⋊ #239 |
| Sylow | C3 ⋊ #713 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C22×D15 |
| commutator, socle, cosocle | C2×C3×C5 | C23 |
| Fitting | C3×C5×Q8 | C2 |
| hypercenter | Q8 | D15 |
| hypocenter | C3×C5 | #41 |
| LCS | G ▷ C2×C3×C5 ▷ C3×C5 | |
| UCS | C1 ◁ C2 ◁ Q8 | |
80 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 120 |
A minimal faithful permutation representation — 3 generators on 16 points:
(1,2,4,5,3)(6,7,8)(9,11,12,15)(10,13,14,16);(9,10,12,14)(11,16,15,13);(2,3)(4,5)(7,8)(11,15)(13,16)Polycyclic presentation:
2^4,3,5|b^2=d,c^2=d,c^a=cd,e^a=e^2,f^a=f^4,c^b=cd45 conjugacy classes — view the full character table.
| maximal subgroup of | #91506, #91512, #91521, … |
| maximal normal subgroup of | #91506, #91512, #91521, … |
| maximal quotient of | #91779, #91781, #91782, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(240, 182);