Finite Group DatabasebetaFinite Group Databasebeta| order | 804 |
| isoclinism | isoclinic class of A4 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C67 · Schur multiplier (H2) = C2 |
| direct product | A4×C67 |
| unambiguous semidirect product | C22⋊(C3×C67) |
| Fitting, socle, Hall | C22×C67 ⋊ C3 |
| derived, Sylow | C22 ⋊ C3×C67 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C67 | A4 |
| commutator, cosocle, hypocenter | C22 | C3×C67 |
| Frattini | C1 | G |
| Fitting, socle | C22×C67 | C3 |
| LCS | G ▷ C22 | |
| UCS | C1 ◁ C67 | |
10 conjugacy classes of subgroups.
| transitive degree | μt(G) = 268 |
268 conjugacy classes. The character table is too large to display.
| derived subgroup of | A4⋊D67 |
| maximal subgroup of | S4×C67, A4⋊D67, A4×D67, … |
| maximal normal subgroup of | S4×C67, A4⋊D67, A4×D67, … |
| maximal quotient of | SL(2,𝔽3)×C67, C2×A4×C67, C22⋊C9×C67, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(804, 10);