Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C23 |
| direct product | C2×C3×#5933 |
| subgroup | G/H | |
|---|---|---|
| center, socle, hypercenter | C22×C3 | #1020 |
| commutator, cosocle, hypocenter | #997 | C22×C3 |
| Frattini | C2 | C2×C3×#1020 |
| Fitting | C2×C3×H(5,𝔽2) | D3 |
| prime radical | C2×C3×#997 | C2 |
| LCS | G ▷ #997 | |
| UCS | C1 ◁ C22×C3 | |
576 conjugacy classes of subgroups; only the 36 normal subgroups are shown.
This group has 576 conjugacy classes of subgroups — too many to draw. Only its 36 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 96 |
78 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,𝔽7)×#5933, C2×C3×#87516, C2×C3×#73030, … |
| maximal normal subgroup of | C2×C3×#87516, C2×C3×#73030, C2×C3×#73027 |
| maximal quotient of | AGL(1,𝔽7)×#5933 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 157765);