Finite Group DatabasebetaFinite Group Databasebeta| order | 10624 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| Sylow | C83 ⋊ #2375 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D83 |
| commutator | C22×C83 | C23×C4 |
| Fitting, prime radical | C2×#120×C83 | C2 |
| socle, cosocle | C23×C83 | C24 |
| hypercenter | C2×#120 | D83 |
| hypocenter | C83 | #2375 |
| LCS | G ▷ C22×C83 ▷ C83 | |
| UCS | C1 ◁ C23 ◁ C2×#120 | |
688 conjugacy classes of subgroups; only the 213 normal subgroups are shown.
This group has 688 conjugacy classes of subgroups — too many to draw. Only its 213 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=g,b^a=be,c^a=cf,d^a=def,h^a=h^82,c^b=cef,d^b=de1186 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #35321169, #35321173, #35321676, … |
| maximal normal subgroup of | #35321169, #35321173, #35321676, … |
| maximal quotient of | #35326022, #35326035, #35326043, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(10624, 7651);