Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| isoclinism | isoclinic class of #63653 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2×C3 |
No aliases for this group.
| Fitting | C32×D8 ⋊ C2 |
| Sylow | C32 ⋊ D16 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #3753 |
| commutator, cosocle | C32×C8 | C22 |
| Frattini | C8 | C2×C3⋊D3 |
| Fitting | C32×D8 | C2 |
| socle | C2×C32 | D8 |
| hypercenter | D8 | C3⋊D3 |
| hypocenter | C32 | D16 |
| LCS | G ▷ C32×C8 ▷ C32×C4 ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ D8 | |
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:
(2,6)(3,10)(4,11)(7,8)(9,15)(12,16)(13,14)(20,21,22);(1,2)(3,6)(4,13)(5,9)(7,10)(8,11)(12,15)(14,16)(18,19)(21,22);(17,18,19)Polycyclic presentation:
2^5,3^2|c^2=de,d^2=e,b^a=bc,c^a=cd,d^a=de,f^a=f^2,g^a=g^2,c^b=cd,d^b=de39 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#826, #10591024, #10591033, … |
| maximal normal subgroup of | D3×#826, #10591024, #10591033, … |
| maximal quotient of | #10589355, #10589357, #10589385, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 301);