Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2×C3 |
No aliases for this group.
| Sylow | C32 ⋊ #125 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C22×C3⋊D3 |
| commutator, socle, prime radical, cosocle | C22×C32 | C23 |
| Fitting | C32×C42 | C2 |
| hypercenter | C42 | C3⋊D3 |
| hypocenter | C32 | #125 |
| LCS | G ▷ C22×C32 ▷ C32 | |
| UCS | C1 ◁ C22 ◁ C42 | |
168 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 288 |
A minimal faithful permutation representation — 3 generators on 22 points:
(4,5,6)(8,12)(11,14)(15,17,18,21)(16,20,19,22);(1,2,3)(7,9,10,13)(8,11,12,14)(16,19)(20,22);(2,3)(5,6)(7,8)(9,11)(10,12)(13,14)(15,16,18,19)(17,20,21,22)Polycyclic presentation:
2^5,3^2|a^2=e,b^2=d,c^2=e,b^a=be,c^a=cd,f^a=f^2,g^a=g^278 conjugacy classes — view the full character table.
| maximal subgroup of | #10589089, #10589331, #10590271, … |
| maximal normal subgroup of | #10589089, #10589331, #10590271, … |
| maximal quotient of | #10592356, #10592359, #10592627, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 727);