Finite Group DatabasebetaFinite Group Databasebeta| order | 8768 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Fitting | M32×C137 ⋊ C2 |
| Sylow | C137 ⋊ #349 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | #14555487 |
| commutator | C22×C137 | C2×C8 |
| Frattini | C2×C8 | C2×D137 |
| Fitting | M32×C137 | C2 |
| socle | C2×C137 | C22⋊C8 |
| hypercenter | M32 | D137 |
| hypocenter | C137 | #349 |
| LCS | G ▷ C22×C137 ▷ C2×C137 ▷ C137 | |
| UCS | C1 ◁ C4 ◁ C2×C8 ◁ M32 | |
58 conjugacy classes of subgroups.
Polycyclic presentation:
2^6,137|a^2=d,d^2=e,e^2=f,b^a=bc,c^a=cf,d^b=df,g^b=g^1361382 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #349×D137, C2×#22577184, #31782408, … |
| maximal normal subgroup of | #349×D137, C2×#22577184, #31782408, … |
| maximal quotient of | #31778712, #31778728, #31778746, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8768, 330);