Finite Group DatabasebetaFinite Group Databasebeta| order | 10624 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C4 |
No aliases for this group.
| Fitting | #322×C83 ⋊ C2 |
| Sylow | C83 ⋊ #1454 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C42 | C22×D83 |
| commutator, socle | C22×C83 | C2×C42 |
| Fitting | #322×C83 | C2 |
| prime radical | C22×D83 | C42 |
| cosocle | C42×C83 | C23 |
| hypercenter | #322 | D83 |
| hypocenter | C83 | #1454 |
| LCS | G ▷ C22×C83 ▷ C83 | |
| UCS | C1 ◁ C42 ◁ #322 | |
178 conjugacy classes of subgroups; only the 89 normal subgroups are shown.
This group has 178 conjugacy classes of subgroups — too many to draw. Only its 89 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Polycyclic presentation:
2^7,83|a^2=d,b^2=e,d^2=f,e^2=g,c^a=cfg,c^b=cf,h^b=h^82,h^c=h^821696 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #1454×D83, C2×#24905764, #35370036, … |
| maximal normal subgroup of | #1454×D83, C2×#24905764, #35370036, … |
| maximal quotient of | #35293778, #35293794, #35294838, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(10624, 2811);