Finite Group DatabasebetaFinite Group Databasebeta| order | 1600 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C5 |
| direct product | C4⋊Q8×C5⋊D5 |
| Fitting | C52×C4⋊Q8 ⋊ C2 |
| Sylow | C52 ⋊ C2×C4⋊Q8 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×C5⋊D5 |
| commutator, socle, cosocle | C22×C52 | C24 |
| Fitting | C52×C4⋊Q8 | C2 |
| prime radical | C22×C5⋊D5 | C23 |
| hypercenter | C4⋊Q8 | C5⋊D5 |
| hypocenter | C52 | C2×C4⋊Q8 |
| LCS | G ▷ C22×C52 ▷ C52 | |
| UCS | C1 ◁ C22 ◁ C4⋊Q8 | |
1160 conjugacy classes of subgroups; only the 287 normal subgroups are shown.
This group has 1160 conjugacy classes of subgroups — too many to draw. Only its 287 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 800 |
196 conjugacy classes — view the full character table.
| maximal subgroup of | C5⋊D5×#492, C5⋊D5×#498, #72220955, … |
| maximal normal subgroup of | C5⋊D5×#492, C5⋊D5×#498, #72220955, … |
| maximal quotient of | C5⋊D5×#498 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1600, 9604);