Finite Group DatabasebetaFinite Group Databasebeta| order | 5312 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C4⋊Q8×C83 ⋊ C2 |
| Sylow | C83 ⋊ #456 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #14581760 |
| commutator | C2×C4×C83 | C23 |
| Frattini | C2×C4 | C22×D83 |
| Fitting | C4⋊Q8×C83 | C2 |
| socle | C2×C83 | #120 |
| hypercenter | C4⋊Q8 | D83 |
| hypocenter | C83 | #456 |
| LCS | G ▷ C2×C4×C83 ▷ C2×C83 ▷ C83 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C4⋊Q8 | |
142 conjugacy classes of subgroups.
Polycyclic presentation:
2^6,83|a^2=f,c^2=f,d^2=f,b^a=bd,c^a=ce,d^a=df,d^b=df,e^b=ef,g^b=g^82,d^c=df590 conjugacy classes — view the full character table.
| maximal subgroup of | #24909484, #24909485, #24909489, … |
| maximal normal subgroup of | #24909484, #24909485, #24909489, … |
| maximal quotient of | #24906424, #24906431, #24906436, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(5312, 731);