Finite Group DatabasebetaFinite Group Databasebeta| order | 1568 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22×C7 |
No aliases for this group.
| Fitting | C72×D8 ⋊ C2 |
| Sylow | C72 ⋊ #135 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | D4×C7⋊D7 |
| commutator, cosocle | C4×C72 | C23 |
| Frattini | C4 | C22×C7⋊D7 |
| Fitting | C72×D8 | C2 |
| socle | C2×C72 | C2×D4 |
| hypercenter | D8 | C7⋊D7 |
| hypocenter | C72 | #135 |
| LCS | G ▷ C4×C72 ▷ C2×C72 ▷ C72 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ D8 | |
310 conjugacy classes of subgroups.
| transitive degree | μt(G) = 784 |
Polycyclic presentation:
2^5,7^2|d^2=e,b^a=be,c^a=ce,f^a=f^6,g^a=g^6,c^b=cd,d^b=de,d^c=de182 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1568, 718);