Finite Group DatabasebetaFinite Group Databasebeta| order | 10624 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2×C8 |
No aliases for this group.
| Fitting | #486×C83 ⋊ C2 |
| Sylow | C83 ⋊ #1721 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | #14581788 |
| commutator, cosocle | C42×C83 | C23 |
| Frattini | C42 | C22×D83 |
| Fitting | #486×C83 | C2 |
| socle | C22×C83 | #127 |
| hypercenter | #486 | D83 |
| hypocenter | C83 | #1721 |
| LCS | G ▷ C42×C83 ▷ C22×C83 ▷ C83 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ #486 | |
242 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 242 conjugacy classes of subgroups — too many to draw. Only its 61 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Polycyclic presentation:
2^7,83|b^2=d,c^2=eg,d^2=f,e^2=g,b^a=bd,c^a=ce,d^a=df,e^a=eg,h^a=h^82,c^b=cf,h^b=h^82928 conjugacy classes — view the full character table.
| maximal subgroup of | #1721×D83, C2×#24907131, #35392195, … |
| maximal normal subgroup of | #1721×D83, C2×#24907131, #35392195, … |
| maximal quotient of | #35298195, #35304206, #35304207, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(10624, 4178);