Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Fitting | C9×C4⋊C4 ⋊ C2 |
| Sylow | C9 ⋊ #126 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C22×D9 |
| commutator, cosocle | C22×C9 | C23 |
| Frattini, socle | C22×C3 | C22×D3 |
| Fitting | C9×C4⋊C4 | C2 |
| prime radical | C22×D9 | C22 |
| hypercenter | C4⋊C4 | D9 |
| hypocenter | C9 | #126 |
| LCS | G ▷ C22×C9 ▷ C9 | |
| UCS | C1 ◁ C22 ◁ C4⋊C4 | |
90 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 144 |
A minimal faithful permutation representation — 3 generators on 25 points:
(1,6,8,7,2,4,3,9,5)(11,15)(14,17)(18,20,21,24)(19,23,22,25);(10,12,13,16)(11,14,15,17)(19,22)(20,24);(2,4)(3,7)(5,6)(8,9)(10,11)(12,14)(13,15)(16,17)(18,19)(20,23)(21,22)(24,25)Polycyclic presentation:
2^5,3^2|b^2=d,c^2=e,f^3=g^2,b^a=be,c^a=cd,f^a=f^2g,g^a=g^2,c^b=ce54 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#63460, #10589615, #10589616, … |
| maximal normal subgroup of | C2×#63460, #10589615, #10589616, … |
| maximal quotient of | #10588331, #10588334, #10588340, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 108);