Finite Group DatabasebetaFinite Group Databasebeta| order | 10624 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| Sylow | C83 ⋊ #1446 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C24 | C22×D83 |
| commutator | C23×C83 | C22×C4 |
| Fitting | C2×#95×C83 | C2 |
| socle, prime radical, cosocle | C24×C83 | C23 |
| hypercenter | C2×#95 | D83 |
| hypocenter | C83 | #1446 |
| LCS | G ▷ C23×C83 ▷ C83 | |
| UCS | C1 ◁ C24 ◁ C2×#95 | |
736 conjugacy classes of subgroups; only the 209 normal subgroups are shown.
This group has 736 conjugacy classes of subgroups — too many to draw. Only its 209 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^2=f,c^2=f,b^a=bd,c^a=ce,c^b=cf,h^c=h^821684 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #35291221, #35296539, #35299525, … |
| maximal normal subgroup of | #35291221, #35296539, #35299525, … |
| maximal quotient of | #35292330, #35292332, #35313310, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(10624, 2780);