Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| automorphism | Aut(G) = C2×D4×S4 (order 384) · Inn(G) = C2×S4 · Out(G) = C23 |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2 |
| direct product | C3×A4⋊Q8 |
| unambiguous semidirect product | A4⋊(C3×Q8) |
90 conjugacy classes of subgroups.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 72 |
A minimal faithful permutation representation — 2 generators on 15 points:
(1,2,4)(5,12,8,7)(6,11,10,9)(13,14,15);(2,3)(5,10,8,6)(7,9,12,11)42 conjugacy classes — view the full character table.
| maximal subgroup of | #10593178, #10593180, #10593185, … |
| maximal normal subgroup of | #10593178, #10593180, #10593185, … |
| maximal quotient of | C3×#5386, C3×#5389, C3×#5391, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 896);