Finite Group DatabasebetaFinite Group Databasebeta| order | 1920 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#87489 |
| Fitting | C22×C5×#41 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C5 | C23×S4 |
| commutator, cosocle | C2×A4 | C24×C5 |
| Frattini | C2 | C23×C5×S4 |
| Fitting | C22×C5×#41 | D3 |
| socle | C23×C5 | C23×D3 |
| hypercenter | C5×#41 | S4 |
| hypocenter | A4 | C5×H(5,𝔽2) |
| LCS | G ▷ C2×A4 ▷ A4 | |
| UCS | C1 ◁ C2×C5 ◁ C5×#41 | |
1720 conjugacy classes of subgroups; only the 204 normal subgroups are shown.
This group has 1720 conjugacy classes of subgroups — too many to draw. Only its 204 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 120 |
A minimal faithful permutation representation — 4 generators on 17 points:
(1,2,3,4,5)(6,7,9)(10,12,13,16)(11,15,14,17);(11,14)(15,17);(10,11)(12,15)(13,14)(16,17);(7,8)(12,16)(15,17)220 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1920, 237121);