Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| isoclinism | isoclinic class of #634 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C3 |
No aliases for this group.
| Sylow | C32 ⋊ M32 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C8 | C2×C3⋊D3 |
| commutator, socle | C2×C32 | C2×C8 |
| Fitting | C2×C32×C8 | C2 |
| prime radical | C22×C32 | C8 |
| cosocle | C32×C8 | C22 |
| hypercenter | C2×C8 | C3⋊D3 |
| hypocenter | C32 | M32 |
| LCS | G ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C8 ◁ C2×C8 | |
78 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)(7,12)(8,13)(14,16)(20,21,22);(1,2,3,7,4,8,9,14,5,6,10,12,11,13,15,16)(18,19)(21,22);(17,18,19)Polycyclic presentation:
2^5,3^2|a^2=c,c^2=d,d^2=e,b^a=be,f^a=f^2,g^a=g^284 conjugacy classes — view the full character table.
| maximal subgroup of | #10589105, #10589109, #10589112, … |
| maximal normal subgroup of | #10589105, #10589109, #10589112, … |
| maximal quotient of | #10589327, C4⋊(C3⋊(C3⋊C16)), #10589415, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 287);