Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| isoclinism | isoclinic class of #3788 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C3 |
No aliases for this group.
| Sylow | C32 ⋊ M32 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C4 | C2×#157 |
| commutator, socle | C2×C32 | C2×C8 |
| Fitting | C32×C8 | C4 |
| prime radical | C2×C3⋊D3 | C8 |
| cosocle | C3⋊(C3⋊C8) | C22 |
| hypercenter | C8 | #157 |
| hypocenter | C32 | M32 |
| LCS | G ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C4 ◁ C8 | |
48 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)(8,13,14,20,15,16,21,22);(1,2)(3,5,6,4)(7,8,10,13,11,14,19,20,12,15,9,16,17,21,18,22)Polycyclic presentation:
2^5,3^2|a^2=c,b^2=d,c^2=d,d^2=e,b^a=be,f^a=fg^2,g^a=f^2g^2,f^c=f^2,g^c=g^236 conjugacy classes — view the full character table.
| maximal subgroup of | #10590013, #10590014, #10590083, … |
| maximal normal subgroup of | #10590013, #10590014, #10590083, … |
| maximal quotient of | #10590175, #10590176, #10590177, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 413);