Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2×C3 |
No aliases for this group.
| 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 |
| prime radical | C3×#41⋊C3 | C2 |
| hypercenter | C4 | C3⋊S4 |
| LCS | G ▷ C3×SL(2,𝔽3) | |
| UCS | C1 ◁ C2 ◁ C4 | |
94 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 35 points:
(1,12)(2,19)(3,17)(4,14)(5,30)(6,29)(7,23)(8,11)(9,32)(10,25)(13,16)(15,28)(18,27)(20,22)(21,31)(24,26)(33,34,35);(1,2,6,10)(3,11,14,7)(4,9,17,21)(5,8,15,23)(12,26,29,20)(13,25,27,19)(16,24,18,22)(28,31,30,32)(34,35);(4,15,18)(5,16,17)(8,21,24)(9,22,23)(12,27,30)(13,28,29)(19,26,32)(20,31,25)Polycyclic presentation:
2^2,3^2,2^3|a^2=g,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 | #10593223, #10593226, #10593232, … |
| maximal normal subgroup of | #10593223, #10593226, #10593232, … |
| maximal quotient of | #10593620, #10593622, C4⋊(C3⋊#226), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 913);