Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C32 ⋊ #99 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #3722 |
| commutator | C22×C32 | C2×C4 |
| Frattini, hypercenter | C23 | D32 |
| Fitting, cosocle | C23×C32 | C22 |
| socle | C2×C32 | C22⋊C4 |
| prime radical | C2×#638 | C2 |
| hypocenter | C32 | #99 |
| LCS | G ▷ C22×C32 ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C2 ◁ C23 | |
131 conjugacy classes of subgroups.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 2 generators on 14 points:
(2,3)(4,5,6)(7,9)(8,11,12,14)(10,13);(1,2,3)(5,6)(7,8)(9,11,13,14)(10,12)Polycyclic presentation:
2^5,3^2|a^2=c,b^2=cd,b^a=bd,d^a=de,g^a=g^2,c^b=ce,d^b=de,f^b=f^239 conjugacy classes — view the full character table.
| maximal subgroup of | #10590078, #10590079, #10590080, … |
| maximal normal subgroup of | #10590078, #10590079, #10590080, … |
| maximal quotient of | #10589146, #10589148, #10589151, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 229);