Finite Group DatabasebetaFinite Group Databasebeta| order | 1600 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| Fitting | C52×C4⋊C4 ⋊ C4 |
| Sylow | C52 ⋊ #387 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C22×#1043 |
| commutator, socle | C22×C52 | C22×C4 |
| Fitting | C52×C4⋊C4 | C4 |
| prime radical, cosocle | C22×C5⋊D5 | C23 |
| hypercenter | C4⋊C4 | #1043 |
| hypocenter | C52 | #387 |
| LCS | G ▷ C22×C52 ▷ C52 | |
| UCS | C1 ◁ C22 ◁ C4⋊C4 | |
616 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 616 conjugacy classes of subgroups — too many to draw. Only its 108 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:
2^6,5^2|a^2=d,b^2=f,c^2=f,c^a=cf,g^a=g^2,h^a=h^2,c^b=ce,g^d=g^4,h^d=h^488 conjugacy classes — view the full character table.
| maximal subgroup of | #72221158, #123×#1043, C22⋊Q8×#1043, … |
| maximal normal subgroup of | #72221158, #123×#1043, C22⋊Q8×#1043, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1600, 7075);