Finite Group DatabasebetaFinite Group Databasebeta| order | 1440 |
| isoclinism | isoclinic class of S6 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C22 |
| general orthogonal | GO−(4,𝔽3) |
| direct product | C2×S6 |
| socle | C2×A6 ⋊ C2 |
| derived | A6 ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, Fitting, hypercenter | C2 | S6 |
| commutator, cosocle, hypocenter | A6 | C22 |
| Frattini | C1 | G |
| socle | C2×A6 | C2 |
| LCS | G ▷ A6 | |
| UCS | C1 ◁ C2 | |
194 conjugacy classes of subgroups.
| transitive degree | μt(G) = 12 |
22 conjugacy classes — view the full character table.
| maximal subgroup of | S8, C2×A8, #72224450, … |
| maximal normal subgroup of | #72224450, #72224455, C22×S6 |
| maximal quotient of | #72218858, #72224449, #72224450, … |
| outer automorphism group of | #585 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1440, 5842);