Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| isoclinism | isoclinic class of #10620215 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| derived, Sylow | C34 ⋊ C22×C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | #10620215 |
| commutator, hypocenter | C34 | C22×C4 |
| Fitting, socle, cosocle | C2×C34 | C23 |
| prime radical | C2×C3×#1113 | C2 |
| LCS | G ▷ C34 | |
| UCS | C1 ◁ C2 | |
774 conjugacy classes of subgroups; only the 101 normal subgroups are shown.
This group has 774 conjugacy classes of subgroups — too many to draw. Only its 101 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 48 |
Polycyclic presentation:
2^4,3^4|c^2=d,e^a=e^2,h^a=h^2,e^b=e^2,f^b=f^2,f^c=f^2,g^c=g^284 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#10620215, #72254914 |
| maximal normal subgroup of | C4×#10620215 |
| maximal quotient of | #72254914 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 3263);