Finite Group DatabasebetaFinite Group Databasebeta| order | 5696 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C22⋊Q8×C89 ⋊ C2 |
| Sylow | C89 ⋊ #554 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D89 |
| commutator, socle | C22×C89 | C24 |
| Fitting | C22⋊Q8×C89 | C2 |
| hypercenter | C22⋊Q8 | D89 |
| hypocenter | C89 | #554 |
| LCS | G ▷ C22×C89 ▷ C89 | |
| UCS | C1 ◁ C22 ◁ C22⋊Q8 | |
228 conjugacy classes of subgroups; only the 105 normal subgroups are shown.
This group has 228 conjugacy classes of subgroups — too many to draw. Only its 105 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Polycyclic presentation:
2^6,89|a^2=e,b^2=ef,d^2=e,b^a=be,c^a=cf,d^b=de,g^d=g^88641 conjugacy classes — view the full character table.
| maximal subgroup of | #554×D89, C2×#19152302, #26033804, … |
| maximal normal subgroup of | #554×D89, C2×#19152302, #26033804, … |
| maximal quotient of | #26023124, #26023126, #26023150, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(5696, 1253);