Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C3 |
No aliases for this group.
| Fitting | C32×#41 ⋊ C2 |
| Sylow | C32 ⋊ H(5,𝔽2) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C23×C3⋊D3 |
| commutator, socle, cosocle | C2×C32 | C24 |
| Fitting | C32×#41 | C2 |
| hypercenter | #41 | C3⋊D3 |
| hypocenter | C32 | H(5,𝔽2) |
| LCS | G ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C2 ◁ #41 | |
498 conjugacy classes of subgroups.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 72 |
A minimal faithful permutation representation — 4 generators on 14 points:
(1,3,4,7)(2,6,5,8)(12,13,14);(2,5)(6,8)(9,10,11);(1,2)(3,6)(4,5)(7,8);(3,7)(6,8)(10,11)(13,14)Polycyclic presentation:
2^5,3^2|d^2=e,d^a=de,f^a=f^2,g^a=g^2,c^b=ce57 conjugacy classes — view the full character table.
| maximal subgroup of | #10591667, #10591690, #10592521, … |
| maximal normal subgroup of | #10591667, #10591690, #10592521, … |
| maximal quotient of | #10595990, #10595992, #10595994, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 1014);