Finite Group DatabasebetaFinite Group Databasebeta| order | 10624 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| Sylow | C83 ⋊ #2820 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D83 |
| commutator, socle, cosocle | C23×C83 | C24 |
| Fitting | C2×#124×C83 | C2 |
| prime radical | C22×D4×C83 | C22 |
| hypercenter | C2×#124 | D83 |
| hypocenter | C83 | #2820 |
| LCS | G ▷ C23×C83 ▷ C83 | |
| UCS | C1 ◁ C23 ◁ C2×#124 | |
464 conjugacy classes of subgroups; only the 133 normal subgroups are shown.
This group has 464 conjugacy classes of subgroups — too many to draw. Only its 133 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=ef,b^2=e,d^2=e,b^a=be,c^a=cf,c^b=cg,d^b=def,h^b=h^82,d^c=de1174 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #35305686, #35305690, #35305705, … |
| maximal normal subgroup of | #35305686, #35305690, #35305705, … |
| maximal quotient of | #35335074, #35335096, #35335115, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(10624, 12561);