Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | Aut(G) = C2×C3×D42 (order 384) · Inn(G) = C2×D4 · Out(G) = C23×C3 |
| homology | Abelianization (H1) = C23×C9 · Schur multiplier (H2) = C22 |
| direct product | C9×AGL(1,ℤ/8) |
| subgroup | G/H | |
|---|---|---|
| center | C2×C9 | C2×D4 |
| commutator | C4 | C23×C9 |
| Frattini, cosocle | C3×C4 | C23×C3 |
| socle | C2×C3 | C2×C3×D4 |
| prime radical | C3×AGL(1,ℤ/8) | C3 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C9 ◁ C2×C4×C9 ◁ G | |
102 conjugacy classes of subgroups.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 72 |
A minimal faithful permutation representation — 3 generators on 17 points:
(1,2)(3,5)(4,6)(7,8)(9,10,12,11,13,15,14,16,17);(2,5)(3,8)(6,7);(2,6)(5,7)99 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 186);