Finite Group DatabasebetaFinite Group Databasebeta| order | 1120 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C7 · Schur multiplier (H2) = C25 |
| direct product | C7×#4149 |
| Fitting | C5×C7×#41 ⋊ C2 |
| Hall | C5×C7 ⋊ H(5,𝔽2) |
| Sylow | C5 ⋊ C7×H(5,𝔽2) |
| subgroup | G/H | |
|---|---|---|
| center | C2×C7 | C23×D5 |
| commutator, cosocle | C2×C5 | C24×C7 |
| Frattini | C2 | C23×C7×D5 |
| Fitting | C5×C7×#41 | C2 |
| socle | C2×C5×C7 | C24 |
| hypercenter | C7×#41 | D5 |
| hypocenter | C5 | C7×H(5,𝔽2) |
| LCS | G ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C2×C7 ◁ C7×#41 | |
332 conjugacy classes of subgroups; only the 170 normal subgroups are shown.
This group has 332 conjugacy classes of subgroups — too many to draw. Only its 170 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 280 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,3,4,5,6,7)(8,9,11,12,10)(13,15,16,19)(14,18,17,20);(14,17)(18,20);(13,14)(15,18)(16,17)(19,20);(9,10)(11,12)(15,19)(18,20)259 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1120, 1053);