Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| isoclinism | isoclinic class of #63626 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2×C3 |
No aliases for this group.
| Fitting | C32×C16 ⋊ C2 |
| Sylow | C32 ⋊ D16 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #3745 |
| commutator, cosocle | C32×C8 | C22 |
| Frattini | C8 | C2×C3⋊D3 |
| Fitting | C32×C16 | C2 |
| socle | C2×C32 | D8 |
| hypercenter | C16 | C3⋊D3 |
| hypocenter | C32 | D16 |
| LCS | G ▷ C32×C8 ▷ C32×C4 ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ C16 | |
84 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 144 |
A minimal faithful permutation representation — 3 generators on 22 points:
(1,2,7,13,4,8,11,14,5,9,15,16,10,12,3,6)(20,21,22);(2,6)(3,7)(4,10)(8,16)(9,14)(11,15)(12,13)(18,19)(21,22);(17,18,19)Polycyclic presentation:
2^5,3^2|b^2=cd,c^2=de,d^2=e,b^a=bc,c^a=cd,d^a=de,f^a=f^2,g^a=g^275 conjugacy classes — view the full character table.
| maximal subgroup of | #10589071, #10589074, #10589314, … |
| maximal normal subgroup of | #10589071, #10589074, #10589314, … |
| maximal quotient of | #10589314, #10589315, #10589316, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 274);