Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C3 |
No aliases for this group.
| Fitting | C32×M16 ⋊ C2 |
| Sylow | C32 ⋊ #131 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C4 | C22×C3⋊D3 |
| commutator, socle | C2×C32 | C22×C4 |
| Fitting | C32×M16 | C2 |
| prime radical | #3828 | C2 |
| cosocle | C32×C4 | C23 |
| hypercenter | M16 | C3⋊D3 |
| hypocenter | C32 | #131 |
| LCS | G ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C4 ◁ M16 | |
186 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 144 |
A minimal faithful permutation representation — 3 generators on 22 points:
(1,3,4,10,5,9,11,15)(2,7,8,14,6,12,13,16)(20,21,22);(2,6)(3,9)(8,13)(10,15)(17,18,19);(1,2)(3,7)(4,8)(5,6)(9,12)(10,14)(11,13)(15,16)(18,19)(21,22)Polycyclic presentation:
2^5,3^2|c^2=d,d^2=e,b^a=be,f^a=f^2,g^a=g^2,c^b=ce60 conjugacy classes — view the full character table.
| maximal subgroup of | #10590187, #10590977, #10590978, … |
| maximal normal subgroup of | #10590187, #10590977, #10590978, … |
| maximal quotient of | #10592400, #10592405, #10592407, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 764);