Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| automorphism | Aut(G) = C22×((C3⋊S4)⋊C3)⋊C2 (order 1728) · Inn(G) = C2×C3⋊S4 · Out(G) = C2×D3 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2×C3 |
No aliases for this group.
| Fitting | C2×C3×Q8 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×C3⋊S4 |
| commutator, cosocle, hypocenter | C3×SL(2,𝔽3) | C22 |
| Fitting | C2×C3×Q8 | D3 |
| socle | C2×C3 | C2×S4 |
| hypercenter | C22 | C3⋊S4 |
| LCS | G ▷ C3×SL(2,𝔽3) | |
| UCS | C1 ◁ C2 ◁ C22 | |
104 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 3 generators on 19 points:
(1,3,5,11)(2,13,6,7)(4,12,9,10)(8,15,14,16)(17,18,19);(1,2)(3,8)(4,7)(5,6)(9,13)(10,16)(11,14)(12,15)(18,19);(3,9,12)(4,10,11)(7,14,16)(8,15,13)Polycyclic presentation:
2^2,3^2,2^3|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 | #10596435, #10596439, D3×#983, … |
| maximal normal subgroup of | #10596435, #10596439, D3×#983, … |
| maximal quotient of | #10593622, #10593623, #10593628, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 912);