Finite Group DatabasebetaFinite Group Databasebeta| order | 5696 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C4⋊Q8×C89 ⋊ C2 |
| Sylow | C89 ⋊ #456 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #15667980 |
| commutator | C2×C4×C89 | C23 |
| Frattini | C2×C4 | C22×D89 |
| Fitting | C4⋊Q8×C89 | C2 |
| socle | C2×C89 | #120 |
| hypercenter | C4⋊Q8 | D89 |
| hypocenter | C89 | #456 |
| LCS | G ▷ C2×C4×C89 ▷ C2×C89 ▷ C89 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C4⋊Q8 | |
142 conjugacy classes of subgroups.
Polycyclic presentation:
2^6,89|a^2=f,c^2=f,d^2=f,b^a=bd,c^a=ce,d^a=df,d^b=df,e^b=ef,g^b=g^88,d^c=df632 conjugacy classes — view the full character table.
| maximal subgroup of | #26018405, #26018407, #26022273, … |
| maximal normal subgroup of | #26018405, #26018407, #26022273, … |
| maximal quotient of | #26019213, #26019220, #26019225, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(5696, 731);