Finite Group DatabasebetaFinite Group Databasebeta| order | 8724 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | (C727⋊C3)⋊C4 |
| unambiguous semidirect product | Dic727⋊C3 |
| derived, Sylow | C727 ⋊ C3×C4 |
| Hall | Dic727 ⋊ C3 |
| Hall | C727⋊C3 ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D727⋊C3 |
| commutator, hypocenter | C727 | C3×C4 |
| Fitting, socle | C2×C727 | C2×C3 |
| LCS | G ▷ C727 | |
| UCS | C1 ◁ C2 | |
12 conjugacy classes of subgroups.
Polycyclic presentation:
2,3,727,2|a^2=d,c^a=c^726,c^b=c^445254 conjugacy classes — view the full character table.
| maximal subgroup of | C3×(C727⋊C3)⋊C4, C3⋊((C727⋊C3)⋊C4) |
| maximal normal subgroup of | C3×(C727⋊C3)⋊C4 |
| maximal quotient of | Dic727⋊C9, C3×(C727⋊C3)⋊C4, C3⋊((C727⋊C3)⋊C4) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8724, 2);