Finite Group DatabasebetaFinite Group Databasebeta| order | 5312 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | 2−1+4×C83 ⋊ C2 |
| Sylow | C83 ⋊ #577 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C22×C22⋊D83 |
| commutator | C4×C83 | C24 |
| Frattini | C4 | C23×D83 |
| Fitting | 2−1+4×C83 | C2 |
| socle | C2×C83 | C22×D4 |
| hypercenter | 2−1+4 | D83 |
| hypocenter | C83 | #577 |
| LCS | G ▷ C4×C83 ▷ C2×C83 ▷ C83 | |
| 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:
2^6,83|a^2=f,e^2=f,b^a=be,e^a=ef,e^b=ef,g^b=g^82,d^c=df719 conjugacy classes — view the full character table.
| maximal subgroup of | #24919225, #24919231, #24919233, … |
| maximal normal subgroup of | #24919225, #24919231, #24919233, … |
| maximal quotient of | #24915986, #24915996, #24916056, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(5312, 1357);