Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| isoclinism | isoclinic class of #63385 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Sylow | C9 ⋊ Q32 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #3673 |
| commutator, cosocle | C8×C9 | C22 |
| Frattini | C3×C8 | C2×D3 |
| Fitting | C9×Q16 | C2 |
| socle, prime radical | C2×C3 | #213 |
| hypercenter | Q16 | D9 |
| hypocenter | C9 | Q32 |
| LCS | G ▷ C8×C9 ▷ C4×C9 ▷ C2×C9 ▷ C9 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ Q16 | |
36 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 288 |
A minimal faithful permutation representation — 2 generators on 41 points:
(1,9,4,7,6,5,3,2,8)(10,12,15,23)(11,16,19,33)(13,25,28,41)(14,30,26,22)(17,35,36,40)(18,37,29,32)(20,38,39,24)(21,34,31,27);(2,4)(3,7)(5,6)(8,9)(10,11,15,19)(12,20,23,39)(13,24,28,38)(14,29,26,18)(16,31,33,21)(17,34,36,27)(22,40,30,35)(25,37,41,32)Polycyclic presentation:
2^5,3^2|a^2=e,b^2=e,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 | #10588595, #10588596, D9×Q32, … |
| maximal normal subgroup of | #10588595, #10588596, D9×Q32, … |
| maximal quotient of | #10588166, #10588169, #10588241, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 36);