Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| isoclinism | isoclinic class of A4⋊S4 |
| automorphism | Aut(G) = #10596770⋊C2 (order 1152) · Inn(G) = G · Out(G) = C22 |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C22×C3 |
| unambiguous semidirect product | A4⋊S4 |
| Fitting, socle | C24 ⋊ C3⋊D3 |
| derived | A42 ⋊ C2 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | A42 | C2 |
| Fitting, socle | C24 | C3⋊D3 |
| LCS | G ▷ A42 | |
| UCS | C1 | |
95 conjugacy classes of subgroups.
| minimal degree | μ(G) = 8 |
| transitive degree | μt(G) = 12 |
A minimal faithful permutation representation — 3 generators on 8 points:
(1,2,4,3)(5,6,8,7);(2,4,3);(6,8,7)Polycyclic presentation:
2,3^2,2^4|b^a=b^2,c^a=c^2,d^a=e,e^a=d,f^a=g,g^a=f,f^b=fg,g^b=f,d^c=de,e^c=d14 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 1026);