Finite Group DatabasebetaFinite Group Databasebeta| order | 240 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | Aut(G) = C2×C4×#1020⋊C2 (order 1536) · Inn(G) = C22 · Out(G) = C22×C4×S4 |
| homology | Abelianization (H1) = C23×C3×C5 · Schur multiplier (H2) = C22 |
| direct product | C2×C3×C5×Q8 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22×C3×C5 | C22 |
| commutator, Frattini, cosocle | C2 | C23×C3×C5 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3×C5 ◁ G | |
76 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 240 |
A minimal faithful permutation representation — 3 generators on 18 points:
(1,2)(11,12,13)(14,15,16,17,18);(3,4,5,6)(7,10,9,8);(3,7,5,9)(4,8,6,10)150 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(240, 187);