Finite Group DatabasebetaFinite Group Databasebeta| order | 1696 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C22×C4×C53 ⋊ C2 |
| Sylow | C53 ⋊ #123 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C22×D53 |
| commutator, socle, cosocle | C22×C53 | C23 |
| Fitting | C22×C4×C53 | C2 |
| prime radical | C2×C22⋊D53 | C2 |
| hypercenter | C22×C4 | D53 |
| hypocenter | C53 | #123 |
| LCS | G ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C22 ◁ C22×C4 | |
78 conjugacy classes of subgroups.
| transitive degree | μt(G) = 848 |
Polycyclic presentation:
2^5,53|c^2=d,b^a=bd,c^a=ce,f^a=f^52430 conjugacy classes — view the full character table.
| maximal subgroup of | #16810812, #123×D53, C2×#13133200, … |
| maximal normal subgroup of | #16810812, #123×D53, C2×#13133200, … |
| maximal quotient of | #16810486, #16810496, #16810502, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1696, 129);