Finite Group DatabasebetaFinite Group Databasebeta| order | 1696 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Fitting | C8×C53 ⋊ C4 |
| Sylow | C53 ⋊ #97 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C4 | C2×#6157 |
| commutator, socle | C2×C53 | C42 |
| Fitting | C8×C53 | C4 |
| prime radical | C2×D53 | C2×C4 |
| cosocle | C4×D53 | C22 |
| hypercenter | C8 | #6157 |
| hypocenter | C53 | #97 |
| LCS | G ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C4 ◁ C8 | |
40 conjugacy classes of subgroups.
| transitive degree | μt(G) = 424 |
Polycyclic presentation:
2^5,53|a^2=c,b^2=d,c^2=e,b^a=be,f^b=f^30,f^d=f^52124 conjugacy classes — view the full character table.
| maximal subgroup of | #16811513, #16811514, C2×#13133271, … |
| maximal normal subgroup of | #16811513, #16811514, C2×#13133271, … |
| maximal quotient of | #16811464, #88324⋊C8, (C2×#6157)⋊C8, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1696, 200);