Finite Group DatabasebetaFinite Group Databasebeta| order | 10624 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| Sylow | C83 ⋊ #2642 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D83 |
| commutator, socle, prime radical, cosocle | C23×C83 | C24 |
| Fitting | C2×C4⋊Q8×C83 | C2 |
| hypercenter | C2×C4⋊Q8 | D83 |
| hypocenter | C83 | #2642 |
| LCS | G ▷ C23×C83 ▷ C83 | |
| UCS | C1 ◁ C23 ◁ C2×C4⋊Q8 | |
416 conjugacy classes of subgroups; only the 149 normal subgroups are shown.
This group has 416 conjugacy classes of subgroups — too many to draw. Only its 149 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=eg,b^2=f,c^2=e,d^2=e,b^a=be,c^a=cf,h^a=h^82,c^b=cg,h^b=h^82,d^c=de1180 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #35305792, #35305796, #35305848, … |
| maximal normal subgroup of | #35305792, #35305796, #35305848, … |
| maximal quotient of | #35330812, #35330938, #35331011, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(10624, 10460);