Finite Group DatabasebetaFinite Group Databasebeta| order | 1920 |
| isoclinism | isoclinic class of #6728 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C4 |
| direct product | C4×GL(2,𝔽5) |
| derived | SL(2,𝔽5) ⋊ C42 |
| subgroup | G/H | |
|---|---|---|
| center, Fitting, hypercenter | C42 | S5 |
| commutator, hypocenter | SL(2,𝔽5) | C42 |
| Frattini | C2×C4 | C2×S5 |
| socle | C22 | C2×A5⋊C4 |
| prime radical, cosocle | C2×#6732 | C22 |
| LCS | G ▷ SL(2,𝔽5) | |
| UCS | C1 ◁ C42 | |
366 conjugacy classes of subgroups; only the 30 normal subgroups are shown.
This group has 366 conjugacy classes of subgroups — too many to draw. Only its 30 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 96 |
96 conjugacy classes — view the full character table.
| maximal subgroup of | D4×GL(2,𝔽5), Q8×GL(2,𝔽5), C8×GL(2,𝔽5), … |
| maximal normal subgroup of | D4×GL(2,𝔽5), Q8×GL(2,𝔽5), C8×GL(2,𝔽5) |
| maximal quotient of | C8×GL(2,𝔽5), AGL(1,𝔽5)×GL(2,𝔽5), C4×AGL(2,𝔽5) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1920, 240616);