Finite Group DatabasebetaFinite Group Databasebeta| order | 1568 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C72 ⋊ #107 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | #87641 |
| commutator | C4×C72 | C2×C4 |
| Frattini, hypercenter | C2×C4 | D72 |
| Fitting, cosocle | C2×C4×C72 | C22 |
| socle, prime radical | C22×C72 | D4 |
| hypocenter | C72 | #107 |
| LCS | G ▷ C4×C72 ▷ C2×C72 ▷ C72 | |
| UCS | C1 ◁ C22 ◁ C2×C4 | |
111 conjugacy classes of subgroups.
| transitive degree | μt(G) = 224 |
Polycyclic presentation:
2^5,7^2|a^2=d,b^2=c,d^2=e,b^a=bd,g^a=g^6,d^b=de,f^b=f^6164 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1568, 233);