Finite Group DatabasebetaFinite Group Databasebeta| order | 560 |
| isoclinism | isoclinic class of D35 |
| automorphism | Aut(G) = AGL(1,𝔽5)×#120×AGL(1,𝔽7) (order 26880) · Inn(G) = D35 · Out(G) = C3×C4×#120 |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C4 |
| direct product | C4×Dic35 |
| derived, Hall | C5×C7 ⋊ C42 |
| Sylow | C7 ⋊ C4×Dic5 |
| Sylow | C5 ⋊ C4×Dic7 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C4 | D35 |
| commutator, hypocenter | C5×C7 | C42 |
| Frattini | C22 | C2×D35 |
| Fitting | C2×C4×C5×C7 | C2 |
| socle, prime radical, cosocle | C22×C5×C7 | C22 |
| LCS | G ▷ C5×C7 | |
| UCS | C1 ◁ C2×C4 | |
60 conjugacy classes of subgroups.
| transitive degree | μt(G) = 560 |
152 conjugacy classes — view the full character table.
| maximal subgroup of | #11764421, #11764424, #11764446, … |
| maximal normal subgroup of | #11764421, #11764424, #11764446, … |
| maximal quotient of | C4×C7⋊#168, #11764556, C8×Dic35, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(560, 71);