Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | Aut(G) = C24×C3×D4 (order 384) · Inn(G) = D4 · Out(G) = C24×C3 |
| homology | Abelianization (H1) = C2×C4×C9 · Schur multiplier (H2) = C22 |
| direct product | C9×#102 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C9 | D4 |
| commutator | C4 | C2×C4×C9 |
| Frattini, cosocle | C2×C3×C4 | C22×C3 |
| socle | C22×C3 | C3×D4 |
| prime radical | C2×C3×D4 | C2×C3 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C9 ◁ C2×C4×C9 ◁ G | |
75 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 144 |
A minimal faithful permutation representation — 2 generators on 21 points:
(1,2,3,4)(5,6)(7,9)(8,11)(10,12)(13,14,16,15,17,19,18,20,21);(6,9)(7,12)(10,11)126 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 52);