Finite Group DatabasebetaFinite Group Databasebeta| order | 1568 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Sylow | C72 ⋊ M32 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C8 | D72 |
| commutator, socle | C2×C72 | C2×C8 |
| Fitting, cosocle | C72×C8 | C22 |
| prime radical | C2×C7⋊D7 | C8 |
| hypocenter | C72 | M32 |
| LCS | G ▷ C2×C72 ▷ C72 | |
| UCS | C1 ◁ C8 | |
85 conjugacy classes of subgroups.
| transitive degree | μt(G) = 112 |
Polycyclic presentation:
2^5,7^2|a^2=c,b^2=c,c^2=d,d^2=e,b^a=be,g^a=g^6,f^b=f^6188 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1568, 202);