Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C23×C3 |
| unambiguous semidirect product | C4⋊(A4⋊S4) |
| Fitting | C24×C4 ⋊ C3⋊D3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×A4⋊S4 |
| commutator, cosocle | C2×A42 | C22 |
| Fitting | C24×C4 | C3⋊D3 |
| socle | C25 | C2×C3⋊D3 |
| hypercenter | C4 | A4⋊S4 |
| hypocenter | A42 | D4 |
| LCS | G ▷ C2×A42 ▷ A42 | |
| UCS | C1 ◁ C2 ◁ C4 | |
598 conjugacy classes of subgroups; only the 36 normal subgroups are shown.
This group has 598 conjugacy classes of subgroups — too many to draw. Only its 36 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 36 |
Polycyclic presentation:
2^3,3^2,2^4|b^2=c,b^a=bc,d^a=d^2,e^a=e^2,f^a=fg,h^a=hi,h^d=i,i^d=hi,f^e=g,g^e=fg44 conjugacy classes — view the full character table.
| maximal subgroup of | D4×A4⋊S4, C2×C4⋊(A4⋊S4), #72225957, … |
| maximal normal subgroup of | D4×A4⋊S4, C2×C4⋊(A4⋊S4), #72225957, … |
| maximal quotient of | C2×C4⋊(A4⋊S4), #72226057, #72226061, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 157520);