Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C2×Q8×C9 ⋊ C2 |
| Sylow | C9 ⋊ C22⋊Q8 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C22×D9 |
| commutator, cosocle | C22×C9 | C23 |
| Frattini, socle | C22×C3 | C22×D3 |
| Fitting | C2×Q8×C9 | C2 |
| prime radical | C22×D9 | C22 |
| hypercenter | C2×Q8 | D9 |
| hypocenter | C9 | C22⋊Q8 |
| LCS | G ▷ C22×C9 ▷ C9 | |
| UCS | C1 ◁ C22 ◁ C2×Q8 | |
111 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 144 |
A minimal faithful permutation representation — 3 generators on 21 points:
(1,6,8,7,2,4,3,9,5)(11,13)(14,16,17,21)(15,18,19,20);(14,15,17,19)(16,20,21,18);(2,4)(3,7)(5,6)(8,9)(10,11)(12,13)Polycyclic presentation:
2^5,3^2|b^2=e,c^2=e,f^3=g^2,c^a=cd,f^a=f^2g,g^a=g^2,c^b=ce54 conjugacy classes — view the full character table.
| maximal subgroup of | #10588485, #10588487, #10588491, … |
| maximal normal subgroup of | #10588485, #10588487, #10588491, … |
| maximal quotient of | #10588652, #10588658, #10588662, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 156);