Finite Group DatabasebetaFinite Group Databasebeta| order | 1920 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#87516 |
| Fitting | C2×C5×H(5,𝔽2) ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | C2×#1020 |
| commutator, cosocle, hypocenter | #997 | C22×C5 |
| Frattini | C2 | C2×C5×#1020 |
| Fitting | C2×C5×H(5,𝔽2) | D3 |
| hypercenter | C22×C5 | #1020 |
| LCS | G ▷ #997 | |
| UCS | C1 ◁ C2×C5 ◁ C22×C5 | |
552 conjugacy classes of subgroups; only the 32 normal subgroups are shown.
This group has 552 conjugacy classes of subgroups — too many to draw. Only its 32 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 80 |
A minimal faithful permutation representation — 2 generators on 21 points:
(1,2,3,4,5)(6,16,9)(7,21,13,11,14,17)(10,18,19)(12,20);(6,7)(8,21,15,14)(9,13)(10,11)(12,18,20,16)(17,19)115 conjugacy classes — view the full character table.
| maximal subgroup of | D5×#87516, C5×#72209772 |
| maximal normal subgroup of | D5×#87516 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1920, 237148);