Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| isoclinism | isoclinic class of PSU(3,𝔽2) |
| automorphism | Aut(G) = C2×#10596418 (order 1152) · Inn(G) = PSU(3,𝔽2) · Out(G) = C2×D4 |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C2×C3 |
No aliases for this group.
| Sylow | C32 ⋊ C4⋊C8 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C4 | PSU(3,𝔽2) |
| commutator | C3⋊D3 | C2×C8 |
| Fitting | C32×C4 | Q8 |
| socle | C2×C32 | C4⋊C4 |
| prime radical | C2×C3⋊D3 | C2×C4 |
| cosocle | C4×C3⋊D3 | C22 |
| hypocenter | C32 | C4⋊C8 |
| LCS | G ▷ C3⋊D3 ▷ C32 | |
| UCS | C1 ◁ C4 | |
46 conjugacy classes of subgroups.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 2 generators on 17 points:
(1,4,5,6)(3,8,7,9)(10,11,12,14,13,15,16,17);(2,4,5,9)(3,7,8,6)Polycyclic presentation:
2^5,3^2|a^2=c,b^2=cd,d^2=e,b^a=bc,f^a=fg^2,g^a=f^2g^2,f^b=g,g^b=f^2,f^c=f^2,g^c=g^224 conjugacy classes — view the full character table.
| maximal subgroup of | GO+(4,𝔽2)⋊C8, #10593155, #10593156, … |
| maximal normal subgroup of | GO+(4,𝔽2)⋊C8, #10593155, #10593156, … |
| maximal quotient of | #10590082, #10590083, #10590088, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 392);