Finite Group DatabasebetaFinite Group Databasebeta| order | 1696 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | C4⋊#88322 |
| Sylow | C53 ⋊ C4⋊C8 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2×C4 | C2×D53 |
| commutator | C2×C53 | C2×C8 |
| Fitting | C42×C53 | C2 |
| socle, prime radical | C22×C53 | C2×C4 |
| cosocle | C2×C4×C53 | C22 |
| hypercenter | C42 | D53 |
| hypocenter | C53 | C4⋊C8 |
| LCS | G ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C2×C4 ◁ C42 | |
38 conjugacy classes of subgroups.
| transitive degree | μt(G) = 1696 |
Polycyclic presentation:
2^5,53|a^2=d,b^2=c,d^2=e,b^a=bc,f^a=f^52436 conjugacy classes — view the full character table.
| maximal subgroup of | D4⋊#88322, #16810352, Q8⋊#88322, … |
| maximal normal subgroup of | D4⋊#88322, #16810352, Q8⋊#88322, … |
| maximal quotient of | #16810374, #16810376, #16810378, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1696, 76);