Finite Group DatabasebetaFinite Group Databasebeta| order | 8724 |
| automorphism | Aut(G) order 6333624 |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C1 |
| dicyclic | Dic2181 |
| unambiguous semidirect product | C3⋊Dic727 |
| unambiguous semidirect product | C727⋊Dic3 |
| derived, Hall | C3×C727 ⋊ C4 |
| Sylow | C727 ⋊ Dic3 |
| Sylow | C3 ⋊ Dic727 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D2181 |
| commutator, hypocenter | C3×C727 | C4 |
| Fitting, socle | C2×C3×C727 | C2 |
| LCS | G ▷ C3×C727 | |
| UCS | C1 ◁ C2 | |
12 conjugacy classes of subgroups.
Polycyclic presentation:
2,3,727,2|a^2=d,b^a=b^2,c^a=c^7262184 conjugacy classes. The character table is too large to display.
| maximal subgroup of | Dic6543, C3⋊((C727⋊C3)⋊C4), C3⋊Dic2181, … |
| maximal normal subgroup of | C3⋊((C727⋊C3)⋊C4), C3×Dic2181, Dic4362 |
| maximal quotient of | Dic6543, C3⋊Dic2181, C3×Dic2181, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8724, 6);