Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| automorphism | Aut(G) = C3×(#1020⋊C3)⋊C2 (order 1728) · Inn(G) = C22⋊A4 · Out(G) = C2×C3×D3 |
| homology | Abelianization (H1) = C9 · Schur multiplier (H2) = C23 |
No aliases for this group.
| derived, Sylow | H(5,𝔽2) ⋊ C9 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, hypercenter | C2×C3 | C22⋊A4 |
| commutator, hypocenter | H(5,𝔽2) | C9 |
| Fitting, prime radical, cosocle | C3×H(5,𝔽2) | C3 |
| LCS | G ▷ H(5,𝔽2) | |
| UCS | C1 ◁ C2×C3 | |
81 conjugacy classes of subgroups.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 72 |
A minimal faithful permutation representation — 3 generators on 17 points:
(2,3,5)(6,7,8)(9,10,12,11,13,15,14,16,17);(2,6)(3,7);(1,5)(2,3)(4,8)(6,7)Polycyclic presentation:
3^2,2^5|a^3=b,c^a=d,d^a=cd,e^a=f,f^a=ef,f^c=fg,e^d=eg33 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 351);