Finite Group DatabasebetaFinite Group Databasebeta| order | 8384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C4×Q8×C131 ⋊ C2 |
| Sylow | C131 ⋊ #529 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D131 |
| commutator, socle | C22×C131 | C24 |
| Frattini | C22 | C23×D131 |
| Fitting | C4×Q8×C131 | C2 |
| hypercenter | C4×Q8 | D131 |
| hypocenter | C131 | #529 |
| LCS | G ▷ C22×C131 ▷ C131 | |
| UCS | C1 ◁ C2×C4 ◁ C4×Q8 | |
234 conjugacy classes of subgroups; only the 101 normal subgroups are shown.
This group has 234 conjugacy classes of subgroups — too many to draw. Only its 101 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Polycyclic presentation:
2^6,131|b^2=f,d^2=e,b^a=be,c^a=cf,g^a=g^130,g^b=g^130,g^c=g^1301328 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #30664368, #30664373, #30664420, … |
| maximal normal subgroup of | #30664368, #30664373, #30664420, … |
| maximal quotient of | #30668357, #30668402, #30668422, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8384, 1086);