Finite Group DatabasebetaFinite Group Databasebeta| order | 1696 |
| isoclinism | isoclinic class of #6157 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C16 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | D53⋊C16 |
| derived, Sylow | C53 ⋊ C2×C16 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C8 | #6157 |
| commutator, hypocenter | C53 | C2×C16 |
| Frattini | C4 | C2×#6157 |
| Fitting | C8×C53 | C4 |
| socle | C2×C53 | C2×C8 |
| prime radical | C2×D53 | C8 |
| cosocle | #88322 | C22 |
| LCS | G ▷ C53 | |
| UCS | C1 ◁ C8 | |
28 conjugacy classes of subgroups.
| transitive degree | μt(G) = 848 |
Polycyclic presentation:
2^5,53|a^2=c,c^2=d,d^2=e,f^a=f^30,f^b=f^52,f^c=f^52136 conjugacy classes — view the full character table.
| maximal subgroup of | C16×#6157, #16811510, #16811535, … |
| maximal normal subgroup of | C16×#6157, #16811510, #16811535, … |
| maximal quotient of | #16811509, #16811518, #16811544, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1696, 219);