Finite Group DatabasebetaFinite Group Databasebeta| order | 10624 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| Fitting | Q82×C83 ⋊ C2 |
| Sylow | C83 ⋊ #3483 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C24×D83 |
| commutator, socle, cosocle | C22×C83 | C25 |
| Fitting | Q82×C83 | C2 |
| hypercenter | Q82 | D83 |
| hypocenter | C83 | #3483 |
| LCS | G ▷ C22×C83 ▷ C83 | |
| UCS | C1 ◁ C22 ◁ Q82 | |
1202 conjugacy classes of subgroups; only the 491 normal subgroups are shown.
This group has 1202 conjugacy classes of subgroups — too many to draw. Only its 491 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=f,c^2=f,d^2=f,e^2=fg,b^a=bf,c^a=cg,h^a=h^82,d^b=df,h^c=h^82,h^e=h^821075 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #35380298, #35380304, #35380342, … |
| maximal normal subgroup of | #35380298, #35380304, #35380342, … |
| maximal quotient of | #35563884, #35564184, #35564256, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(10624, 18137);