Finite Group DatabasebetaFinite Group Databasebeta| order | 5312 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C83 · Schur multiplier (H2) = C25 |
| direct product | C2×C22⋊Q8×C83 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C83 | C23 |
| commutator, Frattini, cosocle | C22 | C24×C83 |
| prime radical | C24×C83 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C83 ◁ G | |
322 conjugacy classes of subgroups; only the 194 normal subgroups are shown.
This group has 322 conjugacy classes of subgroups — too many to draw. Only its 194 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
2324 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C83×#1472, C2×#328×C83, C83×#1508, … |
| maximal normal subgroup of | C83×#1472, C2×#328×C83, C83×#1508, … |
| maximal quotient of | C2×#380×C83, C2×#385×C83, C2×#387×C83, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(5312, 204);