Finite Group DatabasebetaFinite Group Databasebeta| order | 5312 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C83 ⋊ #477 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×C22⋊D83 |
| commutator | C2×C4×C83 | C23 |
| Frattini | C2×C4 | C22×D83 |
| Fitting | C4×Q8×C83 | C2 |
| socle | C22×C83 | C2×D4 |
| hypercenter | C4×Q8 | D83 |
| hypocenter | C83 | #477 |
| LCS | G ▷ C2×C4×C83 ▷ C2×C83 ▷ C83 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ C4×Q8 | |
96 conjugacy classes of subgroups.
Polycyclic presentation:
2^6,83|a^2=df,b^2=d,c^2=f,e^2=f,b^a=bd,c^a=ce,e^a=ef,g^a=g^82,e^c=ef839 conjugacy classes — view the full character table.
| maximal subgroup of | #24906584, #24906586, #24906590, … |
| maximal normal subgroup of | #24906584, #24906586, #24906590, … |
| maximal quotient of | #24907468, #24907717, #24908055, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(5312, 844);