Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| isoclinism | isoclinic class of #625 |
| automorphism | Aut(G) = C25×D32 (order 1152) · Inn(G) = D32 · Out(G) = C25 |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | (C2×D3)⋊(C3⋊C8) |
| Sylow | C32 ⋊ C22⋊C8 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2×C4 | D32 |
| commutator | C2×C32 | C2×C8 |
| Fitting, cosocle | C2×C32×C4 | C22 |
| socle | C22×C32 | C2×C4 |
| prime radical | C22×C3×D3 | C4 |
| hypocenter | C32 | C22⋊C8 |
| LCS | G ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C2×C4 | |
111 conjugacy classes of subgroups; only the 46 normal subgroups are shown.
This group has 111 conjugacy classes of subgroups — too many to draw. Only its 46 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 2 generators on 18 points:
(2,4)(5,6,7,9,8,10,11,12)(14,15)(16,17,18);(1,2)(3,4)(13,14,15)(17,18)Polycyclic presentation:
2^5,3^2|b^2=c,c^2=e,b^a=bd,g^a=g^2,f^b=f^260 conjugacy classes — view the full character table.
| maximal subgroup of | #10590248, #10590253, #10590261, … |
| maximal normal subgroup of | #10590248, #10590253, #10590261, … |
| maximal quotient of | #10589080, #10589081, #10589083, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 204);