Finite Group DatabasebetaFinite Group Databasebeta| order | 5312 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Fitting | M32×C83 ⋊ C2 |
| Sylow | C83 ⋊ #350 |
| subgroup | G/H | |
|---|---|---|
| center | C8 | C22⋊D83 |
| commutator | C4×C83 | C2×C8 |
| Frattini | C2×C8 | C2×D83 |
| Fitting | M32×C83 | C2 |
| socle | C2×C83 | C22⋊C8 |
| hypercenter | M32 | D83 |
| hypocenter | C83 | #350 |
| LCS | G ▷ C4×C83 ▷ C2×C83 ▷ C83 | |
| UCS | C1 ◁ C8 ◁ C2×C8 ◁ M32 | |
56 conjugacy classes of subgroups.
Polycyclic presentation:
2^6,83|a^2=d,c^2=f,d^2=e,e^2=f,b^a=bc,g^a=g^82,c^b=cf,d^b=df,g^b=g^82848 conjugacy classes — view the full character table.
| maximal subgroup of | #350×D83, C2×#19127486, #24909089, … |
| maximal normal subgroup of | #350×D83, C2×#19127486, #24909089, … |
| maximal quotient of | C2×#19127486 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(5312, 334);