Finite Group DatabasebetaFinite Group Databasebeta| order | 10624 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Sylow | C83 ⋊ #2931 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2×C4 | C23×D83 |
| commutator, socle | C22×C83 | C23×C4 |
| Fitting, prime radical | #532×C83 | C2 |
| cosocle | C2×C4×C83 | C24 |
| hypercenter | #532 | D83 |
| hypocenter | C83 | #2931 |
| LCS | G ▷ C22×C83 ▷ C83 | |
| UCS | C1 ◁ C2×C4 ◁ #532 | |
500 conjugacy classes of subgroups; only the 229 normal subgroups are shown.
This group has 500 conjugacy classes of subgroups — too many to draw. Only its 229 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,c^2=f,f^2=g,b^a=be,d^a=dg,h^a=h^82,d^b=dg,h^c=h^821204 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #35295409, #35295410, #35295419, … |
| maximal normal subgroup of | #35295409, #35295410, #35295419, … |
| maximal quotient of | #35374920, #35374928, #35374936, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(10624, 13222);