Finite Group DatabasebetaFinite Group Databasebeta| order | 1920 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#87483 |
| Fitting | C23×C5×Q8 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C5 | C23×S4 |
| commutator, cosocle | C2×A4 | C24×C5 |
| Frattini | C2 | C23×C5×S4 |
| Fitting | C23×C5×Q8 | D3 |
| socle | C23×C5 | C23×D3 |
| hypercenter | C2×C5×Q8 | S4 |
| hypocenter | A4 | C5×2−1+4 |
| LCS | G ▷ C2×A4 ▷ A4 | |
| UCS | C1 ◁ C2×C5 ◁ C2×C5×Q8 | |
1526 conjugacy classes of subgroups; only the 204 normal subgroups are shown.
This group has 1526 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) = 25 |
| transitive degree | μt(G) = 240 |
A minimal faithful permutation representation — 4 generators on 25 points:
(1,2,3,4,5)(6,7,9)(11,17)(15,23)(16,25)(22,24);(10,13,14,21)(11,16,17,25)(12,18,19,20)(15,22,23,24);(10,12,14,19)(11,15,17,23)(13,20,21,18)(16,24,25,22);(7,8)(10,11)(12,15)(13,16)(14,17)(18,22)(19,23)(20,24)(21,25)220 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1920, 237115);