Finite Group DatabasebetaFinite Group Databasebeta| order | 10164 |
| isoclinism | isoclinic class of A4 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C7×C112 · Schur multiplier (H2) = C2×C11 |
| direct product | C7×C112×A4 |
| unambiguous semidirect product | C22⋊(C3×C7×C112) |
| unambiguous semidirect product | (C22×C11)⋊(C3×C7×C11) |
| Fitting, socle, Hall | C22×C7×C112 ⋊ C3 |
| derived, Sylow | C22 ⋊ C3×C7×C112 |
| Hall | C22×C112 ⋊ C3×C7 |
| Hall | C22×C7 ⋊ C3×C112 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C7×C112 | A4 |
| commutator, cosocle, hypocenter | C22 | C3×C7×C112 |
| Frattini | C1 | G |
| Fitting, socle | C22×C7×C112 | C3 |
| LCS | G ▷ C22 | |
| UCS | C1 ◁ C7×C112 | |
140 conjugacy classes of subgroups.
3388 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C7×A4×C112⋊C3, A4×C7⋊(C112⋊C3), C112×A4×ΓL(1,𝔽8), … |
| maximal normal subgroup of | C7×A4×C112⋊C3, A4×C7⋊(C112⋊C3), C112×A4×ΓL(1,𝔽8), … |
| maximal quotient of | C7×C112×C22⋊C9, C3×C7×C112×A4 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(10164, 126);