Finite Group DatabasebetaFinite Group Databasebeta| order | 8384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | #104×C131 ⋊ C2 |
| Sylow | C131 ⋊ #444 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | D4×D131 |
| commutator | C4×C131 | C22×C4 |
| Frattini | C2×C4 | C22×D131 |
| Fitting | #104×C131 | C2 |
| socle | C2×C131 | C4×D4 |
| hypercenter | #104 | D131 |
| hypocenter | C131 | #444 |
| LCS | G ▷ C4×C131 ▷ C2×C131 ▷ C131 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ #104 | |
104 conjugacy classes of subgroups; only the 45 normal subgroups are shown.
This group has 104 conjugacy classes of subgroups — too many to draw. Only its 45 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=e,c^2=e,d^2=f,e^2=f,b^a=bd,c^a=cf,d^a=df,d^b=df,g^c=g^130932 conjugacy classes — view the full character table.
| maximal subgroup of | #444×D131, C2×#22553431, #30674859, … |
| maximal normal subgroup of | #444×D131, C2×#22553431, #30674859, … |
| maximal quotient of | #30664770, #30664776, #30664777, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8384, 668);