Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| isoclinism | isoclinic class of #625 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32×C4 · Schur multiplier (H2) = C2×C3 |
| direct product | C32×#3724 |
| Sylow | C34 ⋊ C4⋊C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22×C32 | D32 |
| commutator | C2×C32 | C2×C32×C4 |
| Frattini | C22 | C32×D32 |
| Fitting, socle, prime radical | C22×C34 | C22 |
| cosocle | C22×C32 | C22×C32 |
| hypocenter | C32 | C32×C4⋊C4 |
| LCS | G ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C22×C32 | |
756 conjugacy classes of subgroups; only the 168 normal subgroups are shown.
This group has 756 conjugacy classes of subgroups — too many to draw. Only its 168 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 144 |
270 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 2725);