Finite Group DatabasebetaFinite Group Databasebeta| order | 10624 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C2×#125×C83 ⋊ C2 |
| Sylow | C83 ⋊ #2851 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D83 |
| commutator, socle, cosocle | C23×C83 | C24 |
| Fitting | C2×#125×C83 | C2 |
| prime radical | C23×D83 | C23 |
| hypercenter | C2×#125 | D83 |
| hypocenter | C83 | #2851 |
| LCS | G ▷ C23×C83 ▷ C83 | |
| UCS | C1 ◁ C23 ◁ C2×#125 | |
556 conjugacy classes of subgroups; only the 169 normal subgroups are shown.
This group has 556 conjugacy classes of subgroups — too many to draw. Only its 169 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=e,b^2=f,c^2=e,b^a=be,c^a=cf,d^a=dg,h^d=h^821192 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #35300006, #35300011, #35300026, … |
| maximal normal subgroup of | #35300006, #35300011, #35300026, … |
| maximal quotient of | #35335569, #35335621, #35335643, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(10624, 12815);