Finite Group DatabasebetaFinite Group Databasebeta| order | 6976 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | D109⋊C32 |
| derived, Sylow | C109 ⋊ C2×C32 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C16 | #89137 |
| commutator, hypocenter | C109 | C2×C32 |
| Frattini | C8 | C2×#89137 |
| Fitting | C16×C109 | C4 |
| socle | C2×C109 | C2×C16 |
| LCS | G ▷ C109 | |
| UCS | C1 ◁ C16 | |
34 conjugacy classes of subgroups.
Polycyclic presentation:
2^6,109|a^2=c,c^2=d,d^2=e,e^2=f,g^a=g^33,g^b=g^108,g^c=g^108496 conjugacy classes — view the full character table.
| maximal subgroup of | C32×#89137, #28378138, #28378181, … |
| maximal normal subgroup of | C32×#89137, #28378138, #28378181, … |
| maximal quotient of | #28378137, #28378146, #28378190, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6976, 1480);