Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | Aut(G) = C23×C3×C22⋊A4 (order 1152) · Inn(G) = C23 · Out(G) = C22×C3×A4 |
| homology | Abelianization (H1) = C23×C9 · Schur multiplier (H2) = C2 |
| direct product | C9×#126 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C9 | C23 |
| commutator | C22 | C23×C9 |
| Frattini, socle, cosocle | C22×C3 | C23×C3 |
| prime radical | C23×C3 | C22×C3 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C9 ◁ G | |
90 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 144 |
A minimal faithful permutation representation — 3 generators on 25 points:
(1,2,4,3,5,7,6,8,9)(10,11)(12,14)(13,15)(16,17)(18,19)(20,24)(21,22)(23,25);(10,12,13,16)(11,14,15,17)(19,22)(23,24);(11,15)(14,17)(18,20,21,25)(19,23,22,24)126 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 176);