Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32 · Schur multiplier (H2) = C23×C3 |
| direct product | D4×A42 |
| unambiguous semidirect product | C4⋊(C2×A42) |
| unambiguous semidirect product | (C22×D4×A4)⋊C3 |
| unambiguous semidirect product | (D4×C22⋊A4)⋊C3 |
| Fitting, Sylow | C24×D4 ⋊ C32 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C22×A42 |
| commutator, socle, cosocle | C25 | C22×C32 |
| Fitting | C24×D4 | C32 |
| hypercenter | D4 | A42 |
| hypocenter | C24 | C32×D4 |
| LCS | G ▷ C25 ▷ C24 | |
| UCS | C1 ◁ C2 ◁ D4 | |
1028 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 1028 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) = 36 |
80 conjugacy classes — view the full character table.
| maximal subgroup of | D4×A4⋊S4, A42×S4, C2×D4×A42, … |
| maximal normal subgroup of | D4×A4⋊S4, C2×D4×A42, A4×#5414, … |
| maximal quotient of | C2×D4×A42, D4×Ω+(4,𝔽3), #72226051, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 157675);