Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| isoclinism | isoclinic class of C3⋊(#41⋊C3) |
| automorphism | Aut(G) = C22×D3×S4 (order 576) · Inn(G) = D3×A4 · Out(G) = C23 |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C3⋊(#131⋊C3) |
| unambiguous semidirect product | #948⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C4 | D3×A4 |
| commutator, hypocenter | C3×Q8 | C3×C4 |
| Fitting, cosocle | C3×#41 | C2×C3 |
| socle | C2×C3 | C4×A4 |
| prime radical | C3×#41⋊C3 | C2 |
| LCS | G ▷ C3×Q8 | |
| UCS | C1 ◁ C4 | |
49 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 2 generators on 35 points:
(1,2,3,7,6,10,13,20)(4,21,12,18,16,24,26,31,5,8,25,19,15,23,28,30,14,9,11,29,17,22,27,32)(34,35);(1,16,6,14)(2,23,10,21)(3,27,13,25)(4,5,15,17)(7,31,20,29)(8,9,22,24)(11,12,26,28)(18,19,30,32)(33,34,35)Polycyclic presentation:
2,3,2^3,3,2|a^2=c,c^2=g,d^2=g,e^2=g,f^a=f^2,d^b=de,e^b=d,e^d=eg42 conjugacy classes — view the full character table.
| maximal subgroup of | #10593219, #10593220, #10593221, … |
| maximal normal subgroup of | #10593219, #10593220, #10593221, … |
| maximal quotient of | C3⋊C8×SL(2,𝔽3), C2×C3⋊(#131⋊C3), #11720752, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 410);