Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| isoclinism | isoclinic class of #95⋊C3 |
| automorphism | Aut(G) = C3×#4626⋊C2 (order 1152) · Inn(G) = C42⋊C3 · Out(G) = C3×D4 |
| homology | Abelianization (H1) = C9 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | #95⋊C9 |
| derived, Sylow | #95 ⋊ C9 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3 | C42⋊C3 |
| commutator, hypocenter | #95 | C9 |
| Frattini, socle, prime radical | C23×C3 | A4 |
| Fitting, cosocle | C3×#95 | C3 |
| LCS | G ▷ #95 | |
| UCS | C1 ◁ C2×C3 | |
37 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 36 |
A minimal faithful permutation representation — 2 generators on 21 points:
(1,2,5)(3,7,12)(4,8,11)(6,10,9)(13,14,16,15,17,19,18,20,21);(1,3,4,9)(5,10)(6,7)(11,12)Polycyclic presentation:
3^2,2^5|a^3=b,c^2=ef,d^2=e,c^a=d,d^a=cdf,e^a=fg,f^a=efg,d^c=dg36 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 3);