Finite Group DatabasebetaFinite Group Databasebeta| order | 664 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | C22⋊D83 |
| Fitting | C22×C83 ⋊ C2 |
| Sylow | C83 ⋊ D4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×D83 |
| commutator, socle, cosocle | C2×C83 | C22 |
| Fitting | C22×C83 | C2 |
| hypercenter | C22 | D83 |
| hypocenter | C83 | D4 |
| LCS | G ▷ C2×C83 ▷ C83 | |
| UCS | C1 ◁ C2 ◁ C22 | |
16 conjugacy classes of subgroups.
| minimal degree | μ(G) = 87 |
| transitive degree | μt(G) = 332 |
Polycyclic presentation:
2^3,83|b^a=bc,d^a=d^82169 conjugacy classes — view the full character table.
| maximal subgroup of | D4×D83, C2×C22⋊D83, #13050589, … |
| maximal normal subgroup of | D4×D83, C2×C22⋊D83, #13050589, … |
| maximal quotient of | (C2×D83)⋊C4, C22⋊Dic83, Dic83⋊C4, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(664, 7);