Finite Group DatabasebetaFinite Group Databasebeta| order | 10624 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C83 ⋊ #2183 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2×C8 | C22×D83 |
| commutator, socle, prime radical | C22×C83 | C22×C8 |
| Fitting | #346×C83 | C2 |
| cosocle | C2×C8×C83 | C23 |
| hypercenter | #346 | D83 |
| hypocenter | C83 | #2183 |
| LCS | G ▷ C22×C83 ▷ C83 | |
| UCS | C1 ◁ C2×C8 ◁ #346 | |
126 conjugacy classes of subgroups; only the 71 normal subgroups are shown.
This group has 126 conjugacy classes of subgroups — too many to draw. Only its 71 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=d,c^2=de,e^2=f,f^2=g,b^a=bd,c^a=cg,h^b=h^821696 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #2183×D83, C2×#24909369, #35557895, … |
| maximal normal subgroup of | #2183×D83, C2×#24909369, #35557895, … |
| maximal quotient of | #35319752, #35319756, #35319760, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(10624, 6416);