Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| Sylow | C32 ⋊ #124 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×GO+(4,𝔽2) |
| commutator, cosocle | C2×C3⋊D3 | C23 |
| Fitting | C32×C4 | D4 |
| socle | C2×C32 | C2×D4 |
| prime radical | #3802 | C2 |
| hypercenter | C4 | GO+(4,𝔽2) |
| hypocenter | C32 | #124 |
| LCS | G ▷ C2×C3⋊D3 ▷ C32 | |
| UCS | C1 ◁ C2 ◁ C4 | |
122 conjugacy classes of subgroups.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 3 generators on 14 points:
(1,3,4)(2,5,6)(7,9,10,13)(8,11,12,14);(3,4)(7,8,10,12)(9,14,13,11);(1,2)(3,6)(4,5)(9,13)(11,14)Polycyclic presentation:
2^5,3^2|b^2=e,c^2=e,b^a=bd,c^a=ce,f^a=f^2,c^b=ce,f^b=g,g^b=f,f^d=f^2,g^d=g^224 conjugacy classes — view the full character table.
| maximal subgroup of | #10593290, #10593291, #10593294, … |
| maximal normal subgroup of | #10593290, #10593291, #10593294, … |
| maximal quotient of | #10593342, #10593343, #10593344, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 869);