Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| isoclinism | isoclinic class of #3672 |
| automorphism | Aut(G) = C22×D4×AGL(1,ℤ/9) (order 1728) · Inn(G) = #611 · Out(G) = C23×C3 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C9×#41 ⋊ C2 |
| Sylow | C9 ⋊ #135 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | #611 |
| commutator, cosocle | C4×C9 | C23 |
| Frattini | C3×C4 | C22×D3 |
| Fitting | C9×#41 | C2 |
| socle | C2×C3 | C2×#67 |
| hypercenter | #41 | D9 |
| hypocenter | C9 | #135 |
| LCS | G ▷ C4×C9 ▷ C2×C9 ▷ C9 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ #41 | |
93 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 144 |
A minimal faithful permutation representation — 3 generators on 25 points:
(1,2,5,7,9,8,3,6,4)(11,16)(13,21)(15,22)(17,18)(19,25)(23,24);(10,12,14,20)(11,15,18,24)(13,19,21,25)(16,22,17,23);(2,4)(3,7)(5,6)(8,9)(10,11)(12,15)(13,16)(14,18)(17,21)(19,22)(20,24)(23,25)Polycyclic presentation:
2^5,3^2|b^2=e,d^2=e,f^3=g^2,c^a=cd,d^a=de,f^a=f^2g,g^a=g^2,d^c=de54 conjugacy classes — view the full character table.
| maximal subgroup of | #10588503, #10588504, #10588824, … |
| maximal normal subgroup of | #10588503, #10588504, #10588824, … |
| maximal quotient of | #10588676, #10588680, #10588683, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 161);