Finite Group DatabasebetaFinite Group Databasebeta| order | 1568 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2×C7 |
No aliases for this group.
| Fitting | C72×D8 ⋊ C2 |
| Sylow | C72 ⋊ D16 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #11716071 |
| commutator, cosocle | C72×C8 | C22 |
| Frattini | C8 | C2×C7⋊D7 |
| Fitting | C72×D8 | C2 |
| socle | C2×C72 | D8 |
| hypercenter | D8 | C7⋊D7 |
| hypocenter | C72 | D16 |
| LCS | G ▷ C72×C8 ▷ C4×C72 ▷ C2×C72 ▷ C72 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ D8 | |
140 conjugacy classes of subgroups.
| transitive degree | μt(G) = 784 |
Polycyclic presentation:
2^5,7^2|c^2=de,d^2=e,b^a=bc,c^a=cd,d^a=de,f^a=f^6,g^a=g^6,c^b=cd,d^b=de179 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1568, 310);