Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| automorphism | Aut(G) = #10596770⋊C2 (order 1152) · Inn(G) = A42 · Out(G) = D4 |
| homology | Abelianization (H1) = C32 · Schur multiplier (H2) = C2×C3 |
| omega | Ω+(4,𝔽3) |
| unambiguous semidirect product | SL(2,𝔽3)⋊A4 |
| unambiguous semidirect product | #994⋊C3 |
| unambiguous semidirect product | #997⋊C3 |
| Fitting, derived, Sylow | H(5,𝔽2) ⋊ C32 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, hypercenter | C2 | A42 |
| commutator, Fitting, cosocle, hypocenter | H(5,𝔽2) | C32 |
| LCS | G ▷ H(5,𝔽2) | |
| UCS | C1 ◁ C2 | |
52 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 2 generators on 24 points:
(1,19,13,21,3,10,8,24,2,4,18,22)(5,15,11,7,14,9,6,17,23,20,16,12);(1,11,18,20,2,24,8,23,3,7,13,19)(4,10,16,6,9,15,21,22,14,5,12,17)Polycyclic presentation:
3^2,2^5|c^2=g,d^2=g,e^2=g,f^2=g,e^a=ef,f^a=e,c^b=cd,d^b=c,d^c=dg,f^e=fg25 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 860);