Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| automorphism | Aut(G) = D4×((C22⋊C9)⋊C2)⋊C3 (order 1728) · Inn(G) = C2×(C22⋊C9)⋊C2 · Out(G) = C22×C3 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | (C22⋊C9)⋊Q8 |
72 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 72 |
A minimal faithful permutation representation — 2 generators on 21 points:
(1,2,4)(5,7,8,12)(6,9,10,11)(13,18,20,19,14,16,15,21,17);(2,3)(5,6,8,10)(7,11,12,9)(14,16)(15,19)(17,18)(20,21)Polycyclic presentation:
2^3,3^2,2^2|a^2=c,b^2=c,d^3=e^2,b^a=bc,d^a=d^2e,e^a=e^2,f^a=g,g^a=f,f^d=g,g^d=fg30 conjugacy classes — view the full character table.
| maximal subgroup of | #10589500, #10589503, #10589516, … |
| maximal normal subgroup of | #10589500, #10589503, #10589516, … |
| maximal quotient of | #10589488, #10589491, #10589493, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 332);