Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C9 ⋊ #131 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | C22×D9 |
| commutator | C2×C9 | C22×C4 |
| Frattini | C3×C4 | C22×D3 |
| Fitting | C9×#41 | C2 |
| socle | C2×C3 | C22×Dic3 |
| prime radical | C3×#41 | D3 |
| cosocle | C4×C9 | C23 |
| hypercenter | #41 | D9 |
| hypocenter | C9 | #131 |
| LCS | G ▷ C2×C9 ▷ C9 | |
| UCS | C1 ◁ C4 ◁ #41 | |
93 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 144 |
A minimal faithful permutation representation — 3 generators on 25 points:
(1,6,8,7,2,4,3,9,5)(12,18)(15,21)(19,24)(22,25);(10,12)(11,15)(13,19)(14,18)(16,22)(17,21)(20,24)(23,25);(2,4)(3,7)(5,6)(8,9)(10,11,13,16,14,17,20,23)(12,15,19,22,18,21,24,25)Polycyclic presentation:
2^5,3^2|a^2=d,d^2=e,f^3=g^2,f^a=f^2g,g^a=g^2,c^b=ce60 conjugacy classes — view the full character table.
| maximal subgroup of | #10588503, #10588504, #10588876, … |
| maximal normal subgroup of | #10588503, #10588504, #10588876, … |
| maximal quotient of | #10588675, #10588677, #10588678, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 158);