Finite Group DatabasebetaFinite Group Databasebeta| order | 1568 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C72×Q8 ⋊ C22 |
| Sylow | C72 ⋊ H(5,𝔽2) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C22×D72 |
| commutator, socle, cosocle | C2×C72 | C24 |
| Fitting | C72×Q8 | C22 |
| hypercenter | Q8 | D72 |
| hypocenter | C72 | H(5,𝔽2) |
| LCS | G ▷ C2×C72 ▷ C72 | |
| UCS | C1 ◁ C2 ◁ Q8 | |
392 conjugacy classes of subgroups.
| transitive degree | μt(G) = 112 |
Polycyclic presentation:
2^5,7^2|c^2=e,d^2=e,d^a=de,f^a=f^6,c^b=ce,g^b=g^6,d^c=de122 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1568, 868);