Finite Group DatabasebetaFinite Group Databasebeta| order | 5696 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C2×M16×C89 ⋊ C2 |
| Sylow | C89 ⋊ #417 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×D356 |
| commutator | C4×C89 | C22×C4 |
| Frattini | C2×C4 | C22×D89 |
| Fitting | C2×M16×C89 | C2 |
| socle | C22×C89 | C2×D4 |
| hypercenter | C2×M16 | D89 |
| hypocenter | C89 | #417 |
| LCS | G ▷ C4×C89 ▷ C2×C89 ▷ C89 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×M16 | |
162 conjugacy classes of subgroups; only the 63 normal subgroups are shown.
This group has 162 conjugacy classes of subgroups — too many to draw. Only its 63 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,d^2=f,b^a=bd,d^a=df,g^a=g^88,c^b=cf,d^b=df,g^b=g^88902 conjugacy classes — view the full character table.
| maximal subgroup of | #26020207, #26020208, #26020212, … |
| maximal normal subgroup of | #26020207, #26020208, #26020212, … |
| maximal quotient of | #26018640, #26018666, #26018692, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(5696, 551);