Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C23 |
| direct product | C2×C3⋊(B(4,𝔽2)⋊C3) |
| unambiguous semidirect product | (C2×C3)⋊(B(4,𝔽2)⋊C3) |
| unambiguous semidirect product | (C2×#5244)⋊C3 |
| Fitting | C2×C3×H(5,𝔽2) ⋊ C2×C3 |
| derived | C3×H(5,𝔽2) ⋊ C22×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | C3⋊(#120⋊C3) |
| commutator, cosocle, hypocenter | C3×H(5,𝔽2) | C22×C3 |
| Frattini | C23 | C2×D3×A4 |
| Fitting | C2×C3×H(5,𝔽2) | C2×C3 |
| socle | C22×C3 | #120⋊C3 |
| LCS | G ▷ C3×H(5,𝔽2) | |
| UCS | C1 ◁ C22 | |
512 conjugacy classes of subgroups; only the 31 normal subgroups are shown.
This group has 512 conjugacy classes of subgroups — too many to draw. Only its 31 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 48 |
54 conjugacy classes — view the full character table.
| maximal subgroup of | #72274870, C2×#11920740, #72274889, … |
| maximal normal subgroup of | #72274870, C2×#11920740, #72274889, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 155387);