Finite Group DatabasebetaFinite Group Databasebeta| order | 8768 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | 2−1+4×C137 ⋊ C2 |
| Sylow | C137 ⋊ #577 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C22×C22⋊D137 |
| commutator | C4×C137 | C24 |
| Frattini | C4 | C23×D137 |
| Fitting | 2−1+4×C137 | C2 |
| socle | C2×C137 | C22×D4 |
| hypercenter | 2−1+4 | D137 |
| hypocenter | C137 | #577 |
| LCS | G ▷ C4×C137 ▷ C2×C137 ▷ C137 | |
| UCS | C1 ◁ C2 ◁ #41 ◁ 2−1+4 | |
258 conjugacy classes of subgroups; only the 107 normal subgroups are shown.
This group has 258 conjugacy classes of subgroups — too many to draw. Only its 107 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,e^2=f,b^a=be,e^a=ef,e^b=ef,g^b=g^136,d^c=df1178 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #31792551, #31792557, #31792559, … |
| maximal normal subgroup of | #31792551, #31792557, #31792559, … |
| maximal quotient of | #31789312, #31789322, #31789382, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8768, 1357);