Finite Group DatabasebetaFinite Group Databasebeta| order | 5312 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C2×C4⋊C4×C83 ⋊ C2 |
| Sylow | C83 ⋊ #537 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D83 |
| commutator, socle | C22×C83 | C24 |
| Fitting | C2×C4⋊C4×C83 | C2 |
| hypercenter | C2×C4⋊C4 | D83 |
| hypocenter | C83 | #537 |
| LCS | G ▷ C22×C83 ▷ C83 | |
| UCS | C1 ◁ C22 ◁ C2×C4⋊C4 | |
294 conjugacy classes of subgroups.
Polycyclic presentation:
2^6,83|d^2=e,b^a=be,c^a=cf,d^a=de,g^a=g^82,c^b=ce,g^c=g^82842 conjugacy classes — view the full character table.
| maximal subgroup of | #24906397, #24906400, #24906431, … |
| maximal normal subgroup of | #24906397, #24906400, #24906431, … |
| maximal quotient of | #24910056, #24910081, #24910146, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(5312, 1130);