Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2×C3 |
No aliases for this group.
| Fitting | C3×#41 ⋊ D3 |
| derived | C3×SL(2,𝔽3) ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×C3⋊S4 |
| commutator, cosocle, hypocenter | C3×SL(2,𝔽3) | C22 |
| Fitting | C3×#41 | D3 |
| socle | C2×C3 | C2×S4 |
| hypercenter | C4 | C3⋊S4 |
| LCS | G ▷ C3×SL(2,𝔽3) | |
| UCS | C1 ◁ C2 ◁ C4 | |
110 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 3 generators on 19 points:
(1,7)(2,11)(3,6)(4,16)(5,13)(8,10)(9,15)(12,14)(17,18,19);(2,6)(3,4)(7,14)(8,13)(9,11)(10,12)(18,19);(3,9,12)(4,10,11)(7,14,16)(8,15,13)Polycyclic presentation:
2^2,3^2,2^3|b^2=g,e^2=g,f^2=g,b^a=bg,c^a=c^2,d^a=d^2,e^a=f,f^a=e,e^c=fg,f^c=ef,f^e=fg27 conjugacy classes — view the full character table.
| maximal subgroup of | #10593224, #10593225, #10593230, … |
| maximal normal subgroup of | #10593224, #10593225, #10593230, … |
| maximal quotient of | #10593625, C4⋊(C3⋊GL(2,𝔽3)), #10593628, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 915);