Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| isoclinism | isoclinic class of #634 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C23×C3 |
No aliases for this group.
| Sylow | C32 ⋊ #95 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C2×C3⋊D3 |
| commutator | C2×C32 | C42 |
| Fitting | C22×C32×C4 | C2 |
| socle, prime radical, cosocle | C23×C32 | C22 |
| hypercenter | C22×C4 | C3⋊D3 |
| hypocenter | C32 | #95 |
| LCS | G ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C23 ◁ C22×C4 | |
228 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 288 |
A minimal faithful permutation representation — 3 generators on 18 points:
(5,6,7,8)(10,12)(16,17,18);(1,2,3,4)(9,10)(11,12)(14,15)(17,18);(13,14,15)Polycyclic presentation:
2^5,3^2|a^2=c,b^2=d,b^a=be,f^a=f^2,g^a=g^284 conjugacy classes — view the full character table.
| maximal subgroup of | #10589150, #10589152, #10589333, … |
| maximal normal subgroup of | #10589150, #10589152, #10589333, … |
| maximal quotient of | #10589389, #10589390, #10589392, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 306);