Finite Group DatabasebetaFinite Group Databasebeta| order | 5312 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | #102×C83 ⋊ C2 |
| Sylow | C83 ⋊ #417 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | D4×D83 |
| commutator | C4×C83 | C22×C4 |
| Frattini | C2×C4 | C22×D83 |
| Fitting | #102×C83 | C2 |
| socle | C22×C83 | C2×D4 |
| hypercenter | #102 | D83 |
| hypocenter | C83 | #417 |
| LCS | G ▷ C4×C83 ▷ C2×C83 ▷ C83 | |
| UCS | C1 ◁ C22 ◁ C2×C4 ◁ #102 | |
162 conjugacy classes of subgroups; only the 55 normal subgroups are shown.
This group has 162 conjugacy classes of subgroups — too many to draw. Only its 55 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Polycyclic presentation:
2^6,83|a^2=e,d^2=f,b^a=bd,d^a=df,c^b=cf,d^b=df,g^c=g^82596 conjugacy classes — view the full character table.
| maximal subgroup of | #417×D83, C2×#19127699, #24915977, … |
| maximal normal subgroup of | #417×D83, C2×#19127699, #24915977, … |
| maximal quotient of | #24907181, #24907290, #24907292, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(5312, 547);