Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22×C3 |
No aliases for this group.
| Fitting | C32×D8 ⋊ C2 |
| Sylow | C32 ⋊ AGL(1,ℤ/8) |
| subgroup | G/H | |
|---|---|---|
| center | C2 | D4×C3⋊D3 |
| commutator, cosocle | C32×C4 | C23 |
| Frattini | C4 | C22×C3⋊D3 |
| Fitting | C32×D8 | C2 |
| socle | C2×C32 | C2×D4 |
| hypercenter | D8 | C3⋊D3 |
| hypocenter | C32 | AGL(1,ℤ/8) |
| LCS | G ▷ C32×C4 ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ D8 | |
204 conjugacy classes of subgroups.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 72 |
A minimal faithful permutation representation — 3 generators on 14 points:
(1,2)(3,8)(4,5)(6,7)(12,13,14);(2,6)(3,7)(5,8)(9,10,11);(2,5)(6,8)(10,11)(13,14)Polycyclic presentation:
2^5,3^2|d^2=e,c^a=ce,f^a=f^2,g^a=g^2,c^b=cd,d^b=de,d^c=de39 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#911, #10594729, #10594732, … |
| maximal normal subgroup of | D3×#911, #10594729, #10594732, … |
| maximal quotient of | #10592456, #10592460, #10592461, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 768);