Finite Group DatabasebetaFinite Group Databasebeta| order | 8768 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C22⋊Q8×C137 ⋊ C2 |
| Sylow | C137 ⋊ #554 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D137 |
| commutator, socle | C22×C137 | C24 |
| Fitting | C22⋊Q8×C137 | C2 |
| hypercenter | C22⋊Q8 | D137 |
| hypocenter | C137 | #554 |
| LCS | G ▷ C22×C137 ▷ C137 | |
| UCS | C1 ◁ C22 ◁ C22⋊Q8 | |
228 conjugacy classes of subgroups; only the 107 normal subgroups are shown.
This group has 228 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=e,b^2=ef,d^2=e,b^a=be,c^a=cf,d^b=de,g^b=g^136,g^c=g^136977 conjugacy classes — view the full character table.
| maximal subgroup of | #31792606, #31792608, #31792658, … |
| maximal normal subgroup of | #31792606, #31792608, #31792658, … |
| maximal quotient of | #31783665, #31783668, #31783685, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8768, 1257);