Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| isoclinism | isoclinic class of #3714 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C32 ⋊ #106 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | #626 |
| commutator | C32×C4 | C2×C4 |
| Frattini, hypercenter | C2×C4 | D32 |
| Fitting, cosocle | C2×C32×C4 | C22 |
| socle, prime radical | C22×C32 | D4 |
| hypocenter | C32 | #106 |
| LCS | G ▷ C32×C4 ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C22 ◁ C2×C4 | |
85 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 2 generators on 18 points:
(5,6,7,9,8,10,11,12)(14,15)(16,17,18);(1,2,3,4)(6,9)(7,11)(10,12)(13,14,15)(17,18)Polycyclic presentation:
2^5,3^2|a^2=d,b^2=c,c^2=e,b^a=bc,c^a=ce,g^a=g^2,f^b=f^248 conjugacy classes — view the full character table.
| maximal subgroup of | #10590268, #10590271, #10590292, … |
| maximal normal subgroup of | #10590268, #10590271, #10590292, … |
| maximal quotient of | #10589097, #10589136, #10589137, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 221);