Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| isoclinism | isoclinic class of #1020 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C24 |
| direct product | C2×C3×#5936 |
| derived | C22⋊A4 ⋊ C2×C3×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22×C3 | #1020 |
| commutator, hypocenter | C22⋊A4 | C2×C3×C4 |
| Frattini | C2 | C2×C3×#1020 |
| Fitting, socle | C26×C3 | D3 |
| prime radical | C22×C3×C22⋊A4 | C2 |
| cosocle | C2×C22⋊A4 | C22×C3 |
| LCS | G ▷ C22⋊A4 | |
| UCS | C1 ◁ C22×C3 | |
1729 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 1729 conjugacy classes of subgroups — too many to draw. Only its 66 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 72 |
120 conjugacy classes — view the full character table.
| maximal subgroup of | C2×C3×#73081, C2×#11920748, C2×#11923661, … |
| maximal normal subgroup of | C2×C3×#73081, C2×#11920748, C2×#11923661, … |
| maximal quotient of | AGL(1,𝔽7)×#5936, C2×C3×#11714270 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 157776);