Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| automorphism | ? |
| homology | Abelianization (H1) = C32×C4 · Schur multiplier (H2) = C2×C3 |
| unambiguous semidirect product | (#131⋊C3)⋊A4 |
| unambiguous semidirect product | #85629⋊C3 |
| unambiguous semidirect product | #85666⋊C3 |
| Fitting, Sylow | #3573 ⋊ C32 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C8 | A42 |
| commutator, hypocenter | H(5,𝔽2) | C32×C4 |
| Frattini | C4 | C2×A42 |
| Fitting | #3573 | C32 |
| socle | C2 | C4×A42 |
| prime radical | (#41⋊C3)⋊A4 | C2 |
| cosocle | #585 | C2×C32 |
| LCS | G ▷ H(5,𝔽2) | |
| UCS | C1 ◁ C8 | |
219 conjugacy classes of subgroups; only the 34 normal subgroups are shown.
This group has 219 conjugacy classes of subgroups — too many to draw. Only its 34 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:
2,3^2,2^6|a^2=d,d^2=i,e^2=i,f^2=i,g^2=i,h^2=i,g^b=gh,h^b=g,e^c=ef,f^c=e,f^e=fi,h^g=hi100 conjugacy classes — view the full character table.
| maximal subgroup of | #72228086, #72228092, #72228093, … |
| maximal normal subgroup of | #72228086, #72228092, #72228093, … |
| maximal quotient of | C2×(#131⋊C3)⋊A4 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 156576);