Finite Group DatabasebetaFinite Group Databasebeta| order | 1696 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C53 ⋊ C22⋊Q8 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C22×D53 |
| commutator, socle, cosocle | C22×C53 | C23 |
| Fitting | C22×C4×C53 | C2 |
| prime radical | C23×C53 | C22 |
| hypercenter | C22×C4 | D53 |
| hypocenter | C53 | C22⋊Q8 |
| LCS | G ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C22 ◁ C22×C4 | |
74 conjugacy classes of subgroups.
| transitive degree | μt(G) = 848 |
Polycyclic presentation:
2^5,53|a^2=d,b^2=d,b^a=bd,c^a=ce,f^a=f^52430 conjugacy classes — view the full character table.
| maximal subgroup of | #16810359, #16810361, #16810774, … |
| maximal normal subgroup of | #16810359, #16810361, #16810774, … |
| maximal quotient of | #16810483, #16810496, #16810507, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1696, 125);