Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| isoclinism | isoclinic class of #63546 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Sylow | C32 ⋊ SD32 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #3714 |
| commutator, Fitting, cosocle | C32×C8 | C22 |
| Frattini, hypercenter | C8 | D32 |
| socle | C2×C32 | D8 |
| prime radical | C3×D24 | C2 |
| hypocenter | C32 | SD32 |
| LCS | G ▷ C32×C8 ▷ C32×C4 ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 | |
61 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 2 generators on 22 points:
(1,2,7,13,12,16,3,6,5,9,10,15,4,8,11,14)(18,19)(20,21,22);(2,6)(3,10)(4,12)(7,11)(8,15)(9,14)(13,16)(17,18,19)(21,22)Polycyclic presentation:
2^5,3^2|b^2=cd,c^2=d,d^2=e,b^a=bc,c^a=cde,d^a=de,g^a=g^2,f^b=f^242 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#800, #10590232, #10590239, … |
| maximal normal subgroup of | D3×#800, #10590232, #10590239, … |
| maximal quotient of | #10589115, #10589119, #10589141, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 196);