Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| automorphism | Aut(G) = #11809017 (order 1152) · Inn(G) = C2×GO+(4,𝔽2) · Out(G) = D4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C32 ⋊ #123 |
132 conjugacy classes of subgroups; only the 25 normal subgroups are shown.
This group has 132 conjugacy classes of subgroups — too many to draw. Only its 25 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 3 generators on 14 points:
(1,3,4)(2,5,6)(8,12)(11,14);(3,4)(7,9,10,13)(8,11,12,14);(1,2)(3,6)(4,5)(7,8)(9,11)(10,12)(13,14)Polycyclic presentation:
2^5,3^2|b^2=e,b^a=bd,c^a=ce,f^a=f^2,f^b=g,g^b=f,f^d=f^2,g^d=g^227 conjugacy classes — view the full character table.
| maximal subgroup of | #10593415, #10593518, #10596494, … |
| maximal normal subgroup of | #10593415, #10593518, #10596494, … |
| maximal quotient of | #10593338, #10593339, #10593343, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 881);