Finite Group DatabasebetaFinite Group Databasebeta| order | 1920 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C5 · Schur multiplier (H2) = C22 |
| direct product | C5×#73000 |
| Fitting | C5×#393 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C5 | #1020⋊C2 |
| commutator, cosocle, hypocenter | C22⋊SL(2,𝔽3) | C22×C5 |
| Frattini | C23 | C2×C5×S4 |
| Fitting | C5×#393 | D3 |
| socle | C23×C5 | C2×S4 |
| LCS | G ▷ C22⋊SL(2,𝔽3) | |
| UCS | C1 ◁ C2×C5 | |
252 conjugacy classes of subgroups; only the 24 normal subgroups are shown.
This group has 252 conjugacy classes of subgroups — too many to draw. Only its 24 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 240 |
A minimal faithful permutation representation — 2 generators on 29 points:
(1,2,3,4,5)(6,10,8,11,9,7)(12,13)(14,24,26)(15,20,29,19,25,28)(16,23,18)(21,22);(8,13)(10,11)(14,15)(16,22)(17,25)(18,19)(20,27)(21,24)(23,29)(26,28)105 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1920, 236422);