Finite Group DatabasebetaFinite Group Databasebeta| order | 800 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C4⋊C4×C25 ⋊ C2 |
| Sylow | C25 ⋊ #123 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C22×D25 |
| commutator, cosocle | C22×C25 | C23 |
| Frattini, socle | C22×C5 | C22×D5 |
| Fitting | C4⋊C4×C25 | C2 |
| prime radical | C2×D100 | C2 |
| hypercenter | C4⋊C4 | D25 |
| hypocenter | C25 | #123 |
| LCS | G ▷ C22×C25 ▷ C25 | |
| UCS | C1 ◁ C22 ◁ C4⋊C4 | |
117 conjugacy classes of subgroups; only the 40 normal subgroups are shown.
This group has 117 conjugacy classes of subgroups — too many to draw. Only its 40 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 400 |
Polycyclic presentation:
2^5,5^2|b^2=d,c^2=e,f^5=g,c^a=cd,f^a=f^4g^4,g^a=g^4,c^b=ce134 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#11716456, #13098831, #13098837, … |
| maximal normal subgroup of | C2×#11716456, #13098831, #13098837, … |
| maximal quotient of | #13097935, #13097940, #13097943, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(800, 115);