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×Q8 | C4 |
| prime radical | #3832 | C2 |
| cosocle | C3⋊Dic3 | C23 |
| hypercenter | Q8 | #157 |
| hypocenter | C32 | #131 |
| LCS | G ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C2 ◁ Q8 | |
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)(7,10,12,11)(8,14,16,15)(9,17,18,19)(13,20,21,22);(7,9,12,18)(8,13,16,21)(10,19,11,17)(14,22,15,20);(1,2)(3,5,6,4)(7,8,11,15,12,16,10,14)(9,13,17,20,18,21,19,22)Polycyclic presentation:
2^5,3^2|a^2=d,b^2=e,c^2=e,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 | #10593271, #10593479, #10593481, … |
| maximal normal subgroup of | #10593271, #10593479, #10593481, … |
| maximal quotient of | #10593748, #10593749, #10593751, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 937);