Finite Group DatabasebetaFinite Group Databasebeta| order | 10624 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C83 ⋊ #2977 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2×C4 | C23×D83 |
| commutator, socle | C22×C83 | C23×C4 |
| Fitting, prime radical | #525×C83 | C2 |
| cosocle | C2×C4×C83 | C24 |
| hypercenter | #525 | D83 |
| hypocenter | C83 | #2977 |
| LCS | G ▷ C22×C83 ▷ C83 | |
| UCS | C1 ◁ C2×C4 ◁ #525 | |
422 conjugacy classes of subgroups; only the 205 normal subgroups are shown.
This group has 422 conjugacy classes of subgroups — too many to draw. Only its 205 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=f,f^2=g,b^a=be,d^a=de,h^a=h^82,c^b=ceg,d^b=dg1198 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #35300877, #35300881, #35300884, … |
| maximal normal subgroup of | #35300877, #35300881, #35300884, … |
| maximal quotient of | #35403106, #35403116, #35403174, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(10624, 13566);