Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C32 ⋊ #101 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #3722 |
| commutator, prime radical | C22×C32 | C2×C4 |
| Frattini, hypercenter | C2×C4 | D32 |
| Fitting, cosocle | C2×C32×C4 | C22 |
| socle | C2×C32 | C22⋊C4 |
| hypocenter | C32 | #101 |
| LCS | G ▷ C22×C32 ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C2 ◁ C2×C4 | |
95 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 2 generators on 22 points:
(2,3)(4,5,6)(7,9,15,16,11,17,10,18)(8,12,13,21,14,20,22,19);(1,2,3)(5,6)(7,8,10,13,11,14,15,22)(9,12,16,19,17,20,18,21)Polycyclic presentation:
2^5,3^2|a^2=c,b^2=cde,c^2=e,b^a=bd,d^a=de,g^a=g^2,c^b=ce,d^b=de,f^b=f^239 conjugacy classes — view the full character table.
| maximal subgroup of | #10590059, #10590060, #10590401, … |
| maximal normal subgroup of | #10590059, #10590060, #10590401, … |
| maximal quotient of | #10589089, #10589094, #10589148, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 209);