Finite Group DatabasebetaFinite Group Databasebeta| order | 10624 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| Fitting | #493×C83 ⋊ C2 |
| Sylow | C83 ⋊ #1687 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | #14581763 |
| commutator, cosocle | C42×C83 | C23 |
| Frattini | C42 | C22×D83 |
| Fitting | #493×C83 | C2 |
| socle | C22×C83 | #121 |
| hypercenter | #493 | D83 |
| hypocenter | C83 | #1687 |
| LCS | G ▷ C42×C83 ▷ C22×C83 ▷ C83 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ #493 | |
198 conjugacy classes of subgroups; only the 55 normal subgroups are shown.
This group has 198 conjugacy classes of subgroups — too many to draw. Only its 55 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=df,d^2=f,e^2=g,b^a=bd,c^a=ce,d^a=df,e^a=eg,c^b=cf,e^c=eg,h^c=h^82925 conjugacy classes — view the full character table.
| maximal subgroup of | #35317881, #35317883, #35317888, … |
| maximal normal subgroup of | #35317881, #35317883, #35317888, … |
| maximal quotient of | #35297495, #35300644, #35302784, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(10624, 3986);