Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C32 ⋊ C22⋊Q8 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C2×D32 |
| commutator, socle, cosocle | C22×C32 | C23 |
| Fitting | C2×C32×C4 | C22 |
| prime radical | C22×C3×D3 | C22 |
| hypercenter | C2×C4 | D32 |
| hypocenter | C32 | C22⋊Q8 |
| LCS | G ▷ C22×C32 ▷ C32 | |
| UCS | C1 ◁ C22 ◁ C2×C4 | |
161 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 161 conjugacy classes of subgroups — too many to draw. Only its 56 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 — 3 generators on 18 points:
(1,2,3)(4,5,6)(11,13,14,18)(12,15,16,17);(2,3)(8,10)(11,12,14,16)(13,17,18,15);(5,6)(7,8)(9,10)Polycyclic presentation:
2^5,3^2|b^2=e,c^2=e,b^a=bd,g^a=g^2,c^b=ce,f^b=f^248 conjugacy classes — view the full character table.
| maximal subgroup of | #10590514, #10590515, #10590516, … |
| maximal normal subgroup of | #10590514, #10590515, #10590516, … |
| maximal quotient of | #10590266, #10590269, #10590270, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 504);