Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22×C3 |
No aliases for this group.
| Sylow | C32 ⋊ C22⋊Q8 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C22×C3⋊D3 |
| commutator, socle, cosocle | C22×C32 | C23 |
| Fitting | C32×C22⋊C4 | C2 |
| prime radical | C23×C32 | C22 |
| hypercenter | C22⋊C4 | C3⋊D3 |
| hypocenter | C32 | C22⋊Q8 |
| LCS | G ▷ C22×C32 ▷ C32 | |
| UCS | C1 ◁ C22 ◁ C22⋊C4 | |
222 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 144 |
A minimal faithful permutation representation — 3 generators on 18 points:
(4,5,6)(8,10)(11,13,14,18)(12,15,16,17);(1,2,3)(7,8)(9,10);(2,3)(5,6)(11,12,14,16)(13,17,18,15)Polycyclic presentation:
2^5,3^2|a^2=d,c^2=d,c^a=cd,f^a=f^2,g^a=g^2,c^b=ce54 conjugacy classes — view the full character table.
| maximal subgroup of | #10594029, #10594031, #10594035, … |
| maximal normal subgroup of | #10594029, #10594031, #10594035, … |
| maximal quotient of | #10592345, #10592346, #10592353, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 735);