Finite Group DatabasebetaFinite Group Databasebeta| order | 1696 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | Q16×C53 ⋊ C2 |
| Sylow | C53 ⋊ #135 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | D4×D53 |
| commutator, cosocle | C4×C53 | C23 |
| Frattini | C4 | C22×D53 |
| Fitting | Q16×C53 | C2 |
| socle | C2×C53 | C2×D4 |
| hypercenter | Q16 | D53 |
| hypocenter | C53 | #135 |
| LCS | G ▷ C4×C53 ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ Q16 | |
62 conjugacy classes of subgroups.
| transitive degree | μt(G) = 848 |
Polycyclic presentation:
2^5,53|c^2=e,d^2=e,b^a=bd,d^a=de,f^a=f^52,d^b=de,f^b=f^52,f^c=f^52196 conjugacy classes — view the full character table.
| maximal subgroup of | #16811054, #16811056, #16811060, … |
| maximal normal subgroup of | #16811054, #16811056, #16811060, … |
| maximal quotient of | #16810618, #16810660, #16810705, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1696, 169);