Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| isoclinism | isoclinic class of GO+(4,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | D32⋊C32 |
| Sylow | C32 ⋊ C22⋊C32 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C16 | GO+(4,𝔽2) |
| commutator | C3⋊D3 | C2×C32 |
| Fitting | C32×C16 | D4 |
| socle | C2×C32 | C22⋊C16 |
| prime radical | C2×D32 | C16 |
| cosocle | C16×C3⋊D3 | C22 |
| hypocenter | C32 | C22⋊C32 |
| LCS | G ▷ C3⋊D3 ▷ C32 | |
| UCS | C1 ◁ C16 | |
148 conjugacy classes of subgroups; only the 31 normal subgroups are shown.
This group has 148 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) = 96 |
Polycyclic presentation:
2^7,3^2|a^2=d,d^2=e,e^2=f,f^2=g,b^a=bc,i^a=hi^2,h^b=hi,i^b=i^2,h^c=h^2,i^c=i^2144 conjugacy classes — view the full character table.
| maximal subgroup of | C32×GO+(4,𝔽2), C3×D32⋊C32 |
| maximal normal subgroup of | C32×GO+(4,𝔽2), C3×D32⋊C32 |
| maximal quotient of | C3×D32⋊C32 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 6708);