Finite Group DatabasebetaFinite Group Databasebeta| order | 1600 |
| isoclinism | isoclinic class of D5×AGL(1,𝔽5) |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C210 |
| direct product | C23×D5×AGL(1,𝔽5) |
| unambiguous semidirect product | D5⋊(C22×C4×D5) |
| unambiguous semidirect product | (C5×AGL(1,𝔽5))⋊C24 |
| unambiguous semidirect product | #1042⋊C24 |
| Fitting, socle | C23×C52 ⋊ C2×C4 |
| derived, Sylow | C52 ⋊ C24×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23 | D5×AGL(1,𝔽5) |
| commutator, hypocenter | C52 | C24×C4 |
| Frattini | C1 | G |
| Fitting, socle | C23×C52 | C2×C4 |
| prime radical | C23×D52 | C2 |
| cosocle | C5×D5 | C25 |
| LCS | G ▷ C52 | |
| UCS | C1 ◁ C23 | |
2791 conjugacy classes of subgroups; only the 882 normal subgroups are shown.
This group has 2791 conjugacy classes of subgroups — too many to draw. Only its 882 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 160 |
160 conjugacy classes — view the full character table.
| maximal subgroup of | C23×AGL(1,𝔽5)2, D5×#65779, #72221610, … |
| maximal normal subgroup of | C23×AGL(1,𝔽5)2, D5×#65779, #72221610, … |
| maximal quotient of | #72221502, D5×#66258, C2×D4×D5×AGL(1,𝔽5) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1600, 10270);