Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C3 |
No aliases for this group.
| Sylow | C32 ⋊ #131 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C22×#157 |
| commutator, socle | C2×C32 | C22×C4 |
| Fitting | C32×D4 | C4 |
| prime radical | #3830 | C2 |
| cosocle | C3⋊Dic3 | C23 |
| hypercenter | D4 | #157 |
| hypocenter | C32 | #131 |
| LCS | G ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C2 ◁ D4 | |
102 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 3 generators on 22 points:
(1,3,6)(2,5,4)(9,15)(12,18)(16,21)(19,22);(7,9)(8,12)(10,16)(11,15)(13,19)(14,18)(17,21)(20,22);(1,2)(3,5,6,4)(7,8,10,13,11,14,17,20)(9,12,16,19,15,18,21,22)Polycyclic presentation:
2^5,3^2|a^2=d,d^2=e,f^a=fg^2,g^a=f^2g^2,c^b=ce,f^d=f^2,g^d=g^230 conjugacy classes — view the full character table.
| maximal subgroup of | #10593269, #10593466, #10593467, … |
| maximal normal subgroup of | #10593269, #10593466, #10593467, … |
| maximal quotient of | #10593738, #10593739, #10593740, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 935);