Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| isoclinism | isoclinic class of #634 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C2×C3 |
No aliases for this group.
| Sylow | C32 ⋊ #97 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2×C4 | C2×C3⋊D3 |
| commutator | C2×C32 | C42 |
| Fitting | C32×C42 | C2 |
| socle, prime radical | C22×C32 | C2×C4 |
| cosocle | C2×C32×C4 | C22 |
| hypercenter | C42 | C3⋊D3 |
| hypocenter | C32 | #97 |
| LCS | G ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C2×C4 ◁ C42 | |
120 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 288 |
A minimal faithful permutation representation — 3 generators on 18 points:
(1,2,3,4)(6,9)(10,12)(16,17,18);(5,6,7,10,8,9,11,12)(14,15)(17,18);(13,14,15)Polycyclic presentation:
2^5,3^2|a^2=c,b^2=d,c^2=e,b^a=be,f^a=f^2,g^a=g^284 conjugacy classes — view the full character table.
| maximal subgroup of | #10589086, #10589088, #10589330, … |
| maximal normal subgroup of | #10589086, #10589088, #10589330, … |
| maximal quotient of | #10589320, #10589332, #10589390, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 278);