Finite Group DatabasebetaFinite Group Databasebeta| order | 1120 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C23 |
| unambiguous semidirect product | (C2×AGL(1,𝔽5))⋊Dic7 |
| Hall | C5×C7 ⋊ #95 |
| Sylow | C7 ⋊ (C2×AGL(1,𝔽5))⋊C4 |
| Sylow | C5 ⋊ C7⋊#95 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C22 | D7×AGL(1,𝔽5) |
| commutator | C2×C5×C7 | C42 |
| Fitting, socle | C22×C5×C7 | C2×C4 |
| prime radical, cosocle | C22×C7×D5 | C22 |
| hypocenter | C5×C7 | #95 |
| LCS | G ▷ C2×C5×C7 ▷ C5×C7 | |
| UCS | C1 ◁ C22 | |
152 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 152 conjugacy classes of subgroups — too many to draw. Only its 56 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 280 |
Polycyclic presentation:
2^5,5,7|a^2=e,b^2=c,b^a=bd,g^a=g^6,f^b=f^2,f^c=f^488 conjugacy classes — view the full character table.
| maximal subgroup of | #72218094, AGL(1,𝔽5)×C22⋊Dic7, #72218209, … |
| maximal normal subgroup of | #72218094, AGL(1,𝔽5)×C22⋊Dic7, #72218209, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1120, 236);