Finite Group DatabasebetaFinite Group Databasebeta| order | 1920 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C5 · Schur multiplier (H2) = C22 |
| direct product | C5×#73002 |
| Fitting | C5×#395 ⋊ 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×#395 | D3 |
| socle | C23×C5 | C2×S4 |
| prime radical | C5×#5931 | C2 |
| LCS | G ▷ C22⋊SL(2,𝔽3) | |
| UCS | C1 ◁ C2×C5 | |
240 conjugacy classes of subgroups; only the 24 normal subgroups are shown.
This group has 240 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)(14,28,23,21,24,29,18,19,25,15,16,27)(17,20,22,26);(8,13)(10,11)(16,22)(17,24)(19,26)(20,28)(23,25)(27,29)105 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1920, 236424);