Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| isoclinism | isoclinic class of Dic18 |
| automorphism | Aut(G) = C22×D4×AGL(1,ℤ/9) (order 1728) · Inn(G) = C2×D9 · Out(G) = C24×C3 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C2×C8×C9 ⋊ C2 |
| Sylow | C9 ⋊ #131 |
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:
(1,6,8,7,2,4,3,9,5)(10,12,13,18,14,19,20,24)(11,16,17,23,15,21,22,25);(11,15)(16,21)(17,22)(23,25);(2,4)(3,7)(5,6)(8,9)(10,11)(12,16)(13,17)(14,15)(18,23)(19,21)(20,22)(24,25)Polycyclic presentation:
2^5,3^2|c^2=d,d^2=e,f^3=g^2,b^a=be,f^a=f^2g,g^a=g^284 conjugacy classes — view the full character table.
| maximal subgroup of | #10588185, #10588192, #10588380, … |
| maximal normal subgroup of | #10588185, #10588192, #10588380, … |
| maximal quotient of | C8×Dic18, #10588359, C8×D36, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 112);