Finite Group DatabasebetaFinite Group Databasebeta| order | 10624 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| Fitting | C23×Q8×C83 ⋊ C2 |
| Sylow | C83 ⋊ #3486 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C24×D83 |
| commutator, socle, cosocle | C22×C83 | C25 |
| Fitting | C23×Q8×C83 | C2 |
| hypercenter | C23×Q8 | D83 |
| hypocenter | C83 | #3486 |
| LCS | G ▷ C22×C83 ▷ C83 | |
| UCS | C1 ◁ C22 ◁ C23×Q8 | |
1492 conjugacy classes of subgroups; only the 557 normal subgroups are shown.
This group has 1492 conjugacy classes of subgroups — too many to draw. Only its 557 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Polycyclic presentation:
2^7,83|b^2=f,d^2=f,b^a=bf,c^a=cg,e^a=ef,h^a=h^82,d^b=df1684 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #35296540, #35296542, #35319946, … |
| maximal normal subgroup of | #35296540, #35296542, #35319946, … |
| maximal quotient of | #35564429, #35564686, #35565342, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(10624, 18161);