Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| isoclinism | isoclinic class of GO+(4,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C215 |
| direct product | C24×GO+(4,𝔽2) |
| unambiguous semidirect product | #157⋊C25 |
| unambiguous semidirect product | D32⋊C25 |
| Fitting, socle | C24×C32 ⋊ D4 |
| Sylow | C32 ⋊ C24×D4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C24 | GO+(4,𝔽2) |
| commutator, cosocle | C3⋊D3 | C26 |
| Frattini | C1 | G |
| Fitting, socle | C24×C32 | D4 |
| hypocenter | C32 | C24×D4 |
| LCS | G ▷ C3⋊D3 ▷ C32 | |
| UCS | C1 ◁ C24 | |
16202 conjugacy classes of subgroups; only the 3199 normal subgroups are shown.
This group has 16202 conjugacy classes of subgroups — too many to draw. Only its 3199 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 96 |
144 conjugacy classes — view the full character table.
| maximal subgroup of | C24×AΓL(1,𝔽9), #120×GO+(4,𝔽2), #72220542 |
| maximal normal subgroup of | C24×AΓL(1,𝔽9), #120×GO+(4,𝔽2), #72220542 |
| maximal quotient of | H(5,𝔽2)×GO+(4,𝔽2) |
| automorphism group of | #63739, #63740 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 153182);