Finite Group DatabasebetaFinite Group Databasebeta| order | 10624 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| Fitting | #381×C83 ⋊ C2 |
| Sylow | C83 ⋊ #2308 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D83 |
| commutator | C22×C83 | C23×C4 |
| Fitting | #381×C83 | C2 |
| socle, cosocle | C23×C83 | C24 |
| prime radical | C23×D332 | C2 |
| hypercenter | #381 | D83 |
| hypocenter | C83 | #2308 |
| LCS | G ▷ C22×C83 ▷ C83 | |
| UCS | C1 ◁ C23 ◁ #381 | |
1160 conjugacy classes of subgroups; only the 229 normal subgroups are shown.
This group has 1160 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=g,b^a=be,c^a=cf,d^a=de,c^b=ce,h^b=h^82,h^c=h^821192 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #35297919, #35297922, #35302120, … |
| maximal normal subgroup of | #35297919, #35297922, #35302120, … |
| maximal quotient of | #35325644, #35325648, #35325700, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(10624, 7087);