Finite Group DatabasebetaFinite Group Databasebeta| order | 1120 |
| isoclinism | isoclinic class of C7⋊AGL(1,𝔽5) |
| automorphism | ? |
| homology | Abelianization (H1) = C32 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C7⋊#3928 |
| derived, Hall | C5×C7 ⋊ C32 |
| Sylow | C7 ⋊ #3928 |
| Sylow | C5 ⋊ C7⋊C32 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C8 | C7⋊AGL(1,𝔽5) |
| commutator, hypocenter | C5×C7 | C32 |
| Fitting | C5×C7×C8 | C4 |
| socle, prime radical | C2×C5×C7 | C16 |
| cosocle | C7×#672 | C2 |
| LCS | G ▷ C5×C7 | |
| UCS | C1 ◁ C8 | |
24 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 1120 |
Polycyclic presentation:
2^5,5,7|a^2=b,b^2=c,c^2=d,d^2=e,f^a=f^2,g^a=g^6,f^b=f^4136 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1120, 6);