Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| Fitting | C42×C9 ⋊ C2 |
| Sylow | C9 ⋊ #124 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C22×D9 |
| commutator, cosocle | C22×C9 | C23 |
| Frattini, socle | C22×C3 | C22×D3 |
| Fitting | C42×C9 | C2 |
| prime radical | C2×D36 | C2 |
| hypercenter | C42 | D9 |
| hypocenter | C9 | #124 |
| LCS | G ▷ C22×C9 ▷ C9 | |
| UCS | C1 ◁ C22 ◁ C42 | |
114 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 144 |
A minimal faithful permutation representation — 3 generators on 21 points:
(1,6,8,7,2,4,3,9,5)(10,12,13,11)(15,19)(18,21);(10,11,13,12)(14,16,17,20)(15,18,19,21);(2,4)(3,7)(5,6)(8,9)(11,12)(14,15)(16,18)(17,19)(20,21)Polycyclic presentation:
2^5,3^2|b^2=d,c^2=e,f^3=g^2,b^a=be,c^a=cd,f^a=f^2g,g^a=g^278 conjugacy classes — view the full character table.
| maximal subgroup of | #10588142, #10588378, #10588379, … |
| maximal normal subgroup of | #10588142, #10588378, #10588379, … |
| maximal quotient of | #10588339, #10588352, #10588353, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 85);