Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| isoclinism | isoclinic class of #157 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C16 · Schur multiplier (H2) = C2×C3 |
No aliases for this group.
| derived, Sylow | C32 ⋊ C2×C16 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C8 | #157 |
| commutator, hypocenter | C32 | C2×C16 |
| Frattini | C4 | C2×#157 |
| Fitting | C32×C8 | C4 |
| socle | C2×C32 | C2×C8 |
| prime radical | C2×C3⋊D3 | C8 |
| cosocle | C3⋊(C3⋊C8) | C22 |
| LCS | G ▷ C32 | |
| UCS | C1 ◁ C8 | |
50 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 2 generators on 22 points:
(1,3,6)(2,5,4)(7,9,11,18,12,10,17,19)(8,13,15,21,16,14,20,22);(1,2)(3,5,6,4)(7,8,10,14,11,15,19,22,12,16,9,13,17,20,18,21)Polycyclic presentation:
2^5,3^2|a^2=c,b^2=d,c^2=d,d^2=e,f^a=fg^2,g^a=f^2g^2,f^c=f^2,g^c=g^248 conjugacy classes — view the full character table.
| maximal subgroup of | D32⋊C16, #10590015, #10590082, … |
| maximal normal subgroup of | D32⋊C16, #10590015, #10590082, … |
| maximal quotient of | #10590132, #10590133, #10590174, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 412);