Finite Group DatabasebetaFinite Group Databasebeta| order | 8768 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | #99×C137 ⋊ C2 |
| Sylow | C137 ⋊ B(4,𝔽2) |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #17984058 |
| commutator | C23×C137 | C23 |
| Frattini | C23 | C22×D137 |
| Fitting | #99×C137 | C2 |
| socle | C2×C137 | #120 |
| hypercenter | #99 | D137 |
| hypocenter | C137 | B(4,𝔽2) |
| LCS | G ▷ C23×C137 ▷ C2×C137 ▷ C137 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ #99 | |
198 conjugacy classes of subgroups; only the 39 normal subgroups are shown.
This group has 198 conjugacy classes of subgroups — too many to draw. Only its 39 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Polycyclic presentation:
2^6,137|b^a=bd,c^a=ce,g^a=g^136,e^b=ef,d^c=df,g^c=g^136764 conjugacy classes — view the full character table.
| maximal subgroup of | #31782774, #31782775, #31782780, … |
| maximal normal subgroup of | #31782774, #31782775, #31782780, … |
| maximal quotient of | #31779708, #31779709, #31779713, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8768, 737);