Finite Group DatabasebetaFinite Group Databasebeta| order | 560 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C22×C5×C7 ⋊ C22 |
| Hall | C5×C7 ⋊ C2×D4 |
| Sylow | C7 ⋊ D4×D5 |
| Sylow | C5 ⋊ D4×D7 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×D5×D7 |
| commutator, socle, cosocle | C2×C5×C7 | C23 |
| Fitting | C22×C5×C7 | C22 |
| hypercenter | C22 | D5×D7 |
| hypocenter | C5×C7 | C2×D4 |
| LCS | G ▷ C2×C5×C7 ▷ C5×C7 | |
| UCS | C1 ◁ C2 ◁ C22 | |
108 conjugacy classes of subgroups.
| transitive degree | μt(G) = 140 |
Polycyclic presentation:
2^4,5,7|c^a=cd,f^a=f^6,c^b=cd,e^b=e^459 conjugacy classes — view the full character table.
| maximal subgroup of | #11765387, #11765390, D4×D5×D7, … |
| maximal normal subgroup of | #11765387, #11765390, D4×D5×D7, … |
| maximal quotient of | #11764833, #11764845, #11764855, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(560, 135);