Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C32 ⋊ C2×M16 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C4 | C2×D32 |
| commutator, socle | C2×C32 | C22×C4 |
| Fitting | C2×C32×C4 | C22 |
| prime radical | C22×C3⋊D3 | C4 |
| cosocle | C32×C4 | C23 |
| hypercenter | C2×C4 | D32 |
| hypocenter | C32 | C2×M16 |
| LCS | G ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C4 ◁ C2×C4 | |
163 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 163 conjugacy classes of subgroups — too many to draw. Only its 54 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 3 generators on 14 points:
(2,3)(7,8)(9,10,11)(12,13,14);(1,3,4,8,5,2,6,7)(10,11);(1,2,4,7,5,3,6,8)(13,14)Polycyclic presentation:
2^5,3^2|a^2=d,b^2=d,d^2=e,c^a=ce,g^a=g^2,c^b=ce,f^b=f^248 conjugacy classes — view the full character table.
| maximal subgroup of | #10590045, #10590046, #10590048, … |
| maximal normal subgroup of | #10590045, #10590046, #10590048, … |
| maximal quotient of | #10590247, #10590255, #10590256, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 466);