Finite Group DatabasebetaFinite Group Databasebeta| order | 8768 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| Sylow | C137 ⋊ #390 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C22×D137 |
| commutator | C22×C137 | C22×C4 |
| Fitting | C22×D4×C137 | C2 |
| socle | C23×C137 | C23 |
| hypercenter | C22×D4 | D137 |
| hypocenter | C137 | #390 |
| LCS | G ▷ C22×C137 ▷ C137 | |
| UCS | C1 ◁ C23 ◁ C22×D4 | |
258 conjugacy classes of subgroups; only the 91 normal subgroups are shown.
This group has 258 conjugacy classes of subgroups — too many to draw. Only its 91 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Polycyclic presentation:
2^6,137|a^2=f,b^2=df,c^2=e,b^a=bd,c^a=ce,g^a=g^136,g^b=g^1361388 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #31778913, #31780743, #31780745, … |
| maximal normal subgroup of | #31778913, #31780743, #31780745, … |
| maximal quotient of | #31779018, #31779026, #31779030, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8768, 452);