Finite Group DatabasebetaFinite Group Databasebeta| order | 240 |
| isoclinism | isoclinic class of #6652 |
| automorphism | Aut(G) = C2×D3×D4×AGL(1,𝔽5) (order 1920) · Inn(G) = #1201 · Out(G) = C22×C4 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Hall | C3×C5 ⋊ Q16 |
| Sylow | C5 ⋊ #216 |
| Sylow | C3 ⋊ #689 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #1201 |
| commutator, Fitting, cosocle | C3×C4×C5 | C22 |
| Frattini, hypercenter | C4 | D3×D5 |
| socle, prime radical | C2×C3×C5 | D4 |
| hypocenter | C3×C5 | Q16 |
| LCS | G ▷ C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5 | |
| UCS | C1 ◁ C2 ◁ C4 | |
36 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 240 |
A minimal faithful permutation representation — 2 generators on 24 points:
(1,2,4,5,3)(7,8)(9,11,13,19)(10,14,16,23)(12,21,22,18)(15,24,20,17);(2,3)(4,5)(6,7,8)(9,10,13,16)(11,17,19,24)(12,20,22,15)(14,21,23,18)Polycyclic presentation:
2^4,3,5|a^2=d,b^2=d,c^2=d,b^a=bc,c^a=cd,f^a=f^4,c^b=cd,e^b=e^230 conjugacy classes — view the full character table.
| maximal subgroup of | #91264, #91276, #91278, … |
| maximal normal subgroup of | #91264, #91276, #91278, … |
| maximal quotient of | #90970, #90972, #90989, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(240, 22);