Finite Group DatabasebetaFinite Group Databasebeta| order | 8384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| Sylow | C131 ⋊ #387 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C22×D131 |
| commutator | C22×C131 | C22×C4 |
| Fitting | C2×C42×C131 | C2 |
| socle | C23×C131 | C23 |
| hypercenter | C2×C42 | D131 |
| hypocenter | C131 | #387 |
| LCS | G ▷ C22×C131 ▷ C131 | |
| UCS | C1 ◁ C23 ◁ C2×C42 | |
154 conjugacy classes of subgroups; only the 71 normal subgroups are shown.
This group has 154 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:
2^6,131|a^2=f,b^2=f,c^2=d,b^a=bd,c^a=ce,g^a=g^1302108 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #387×D131, C2×#22553196, #30668403, … |
| maximal normal subgroup of | #387×D131, C2×#22553196, #30668403, … |
| maximal quotient of | #30664188, #30664194, #30664212, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8384, 433);