Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C2×Q8×C9 ⋊ C2 |
| Sylow | C9 ⋊ 2−1+4 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C23×D9 |
| commutator, cosocle | C2×C9 | C24 |
| Frattini, socle | C2×C3 | C23×D3 |
| Fitting | C2×Q8×C9 | C2 |
| hypercenter | C2×Q8 | D9 |
| hypocenter | C9 | 2−1+4 |
| LCS | G ▷ C2×C9 ▷ C9 | |
| UCS | C1 ◁ C2 ◁ C2×Q8 | |
219 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 144 |
A minimal faithful permutation representation — 4 generators on 25 points:
(1,6,8,7,2,4,3,9,5)(11,17)(15,23)(16,25)(22,24);(10,13,14,21)(11,16,17,25)(12,18,19,20)(15,22,23,24);(10,12,14,19)(11,15,17,23)(13,20,21,18)(16,24,25,22);(2,4)(3,7)(5,6)(8,9)(10,11)(12,15)(13,16)(14,17)(18,22)(19,23)(20,24)(21,25)Polycyclic presentation:
2^5,3^2|b^2=e,c^2=e,f^3=g^2,d^a=de,f^a=f^2g,g^a=g^2,c^b=ce57 conjugacy classes — view the full character table.
| maximal subgroup of | #10588432, #10588433, #10588742, … |
| maximal normal subgroup of | #10588432, #10588433, #10588742, … |
| maximal quotient of | #10589602, #10589606, #10589609, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 361);