Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2×C3 |
No aliases for this group.
| Sylow | C32 ⋊ #125 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×PSU(3,𝔽2) |
| commutator, prime radical, cosocle | C2×C3⋊D3 | C23 |
| Fitting | C32×C4 | Q8 |
| socle | C2×C32 | C2×Q8 |
| hypercenter | C4 | PSU(3,𝔽2) |
| hypocenter | C32 | #125 |
| LCS | G ▷ C2×C3⋊D3 ▷ C32 | |
| UCS | C1 ◁ C2 ◁ C4 | |
64 conjugacy classes of subgroups.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 3 generators on 17 points:
(1,4,9)(2,7,3)(5,8,6)(10,12,13,17)(11,15,14,16);(2,3,5,8)(4,6,9,7)(11,14)(15,16);(2,4,5,9)(3,7,8,6)(10,11,13,14)(12,16,17,15)Polycyclic presentation:
2^5,3^2|a^2=d,b^2=de,c^2=e,b^a=bd,c^a=ce,f^a=fg^2,g^a=f^2g^2,f^b=g,g^b=f^2,f^d=f^2,g^d=g^218 conjugacy classes — view the full character table.
| maximal subgroup of | #10593533, #10593534, #10593537, … |
| maximal normal subgroup of | #10593533, #10593534, #10593537, … |
| maximal quotient of | #10593545, #10593550, #10593551, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 891);