Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C32 ⋊ C4⋊Q8 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×GO+(4,𝔽2) |
| commutator, prime radical, cosocle | C2×C3⋊D3 | C23 |
| Fitting | C32×C4 | D4 |
| socle | C2×C32 | C2×D4 |
| hypercenter | C4 | GO+(4,𝔽2) |
| hypocenter | C32 | C4⋊Q8 |
| LCS | G ▷ C2×C3⋊D3 ▷ C32 | |
| UCS | C1 ◁ C2 ◁ C4 | |
106 conjugacy classes of subgroups.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 3 generators on 14 points:
(1,3,4)(2,5,6)(7,10,11,14)(8,12,9,13);(3,4)(7,9,11,8)(10,12,14,13);(1,2)(3,6)(4,5)(7,8,11,9)(10,13,14,12)Polycyclic presentation:
2^5,3^2|a^2=e,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 | #10593156, #10593291, #10593292, … |
| maximal normal subgroup of | #10593156, #10593291, #10593292, … |
| maximal quotient of | #10593339, #10593342, #10593383, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 870);