Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| isoclinism | isoclinic class of #63385 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Fitting | C9×D8 ⋊ C2 |
| Sylow | C9 ⋊ D16 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #3673 |
| commutator, cosocle | C8×C9 | C22 |
| Frattini | C3×C8 | C2×D3 |
| Fitting | C9×D8 | C2 |
| socle | C2×C3 | #213 |
| hypercenter | D8 | D9 |
| hypocenter | C9 | D16 |
| LCS | G ▷ C8×C9 ▷ C4×C9 ▷ C2×C9 ▷ C9 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ D8 | |
42 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 144 |
A minimal faithful permutation representation — 2 generators on 25 points:
(1,9,4,7,6,5,3,2,8)(11,15)(12,19)(13,20)(16,17)(18,24)(21,25)(22,23);(2,4)(3,7)(5,6)(8,9)(10,11)(12,15)(13,22)(14,18)(16,19)(17,20)(21,24)(23,25)Polycyclic presentation:
2^5,3^2|c^2=de,d^2=e,f^3=g^2,b^a=bc,c^a=cd,d^a=de,f^a=f^2g,g^a=g^2,c^b=cd,d^b=de39 conjugacy classes — view the full character table.
| maximal subgroup of | D9×D16, #10588591, #10588594, … |
| maximal normal subgroup of | D9×D16, #10588591, #10588594, … |
| maximal quotient of | #10588166, #10588168, #10588196, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 33);