Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| isoclinism | isoclinic class of A4⋊#1113 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C22×C3 |
| unambiguous semidirect product | A4⋊#1113 |
| Fitting, socle | C22×C33 ⋊ C2×D3 |
| derived | C33×A4 ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C33×A4 | C22 |
| Fitting, socle | C22×C33 | C2×D3 |
| LCS | G ▷ C33×A4 | |
| UCS | C1 | |
574 conjugacy classes of subgroups; only the 45 normal subgroups are shown.
This group has 574 conjugacy classes of subgroups — too many to draw. Only its 45 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:
2^2,3^4,2^2|c^a=c^2,d^a=d^2,e^a=e^2,g^a=gh,d^b=d^2,f^b=f^2,g^c=h,h^c=gh48 conjugacy classes — view the full character table.
| maximal subgroup of | #72220714, #72220734, #72220748, … |
| maximal normal subgroup of | D32×C3⋊S4, D3×#89101, S4×#1113 |
| maximal quotient of | #72220714, #72220734, #72220748, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 3553);