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 | C4 | C22×C3⋊D3 |
| commutator, socle | C2×C32 | C22×C4 |
| Fitting, prime radical | C32×#41 | C2 |
| cosocle | C32×C4 | C23 |
| hypercenter | #41 | C3⋊D3 |
| hypocenter | C32 | #131 |
| LCS | G ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C4 ◁ #41 | |
186 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 144 |
A minimal faithful permutation representation — 3 generators on 22 points:
(3,9)(6,12)(10,15)(13,16)(20,21,22);(1,3)(2,6)(4,10)(5,9)(7,13)(8,12)(11,15)(14,16)(17,18,19);(1,2,4,7,5,8,11,14)(3,6,10,13,9,12,15,16)(18,19)(21,22)Polycyclic presentation:
2^5,3^2|a^2=d,d^2=e,f^a=f^2,g^a=g^2,c^b=ce60 conjugacy classes — view the full character table.
| maximal subgroup of | #10590221, #10591675, #10591676, … |
| maximal normal subgroup of | #10590221, #10591675, #10591676, … |
| maximal quotient of | #10592696, #10592698, #10592699, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 805);