Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| automorphism | Aut(G) = C2×D4×GO+(4,𝔽2) (order 1152) · Inn(G) = D32⋊C4 · Out(G) = C23 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C32 ⋊ #101 |
67 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 2 generators on 22 points:
(1,3)(2,5,6)(7,9,10,17,11,16,19,22)(8,12,13,20,14,15,18,21);(1,2)(3,6)(4,5)(7,8,11,14)(9,15,16,12)(10,18,19,13)(17,20,22,21)Polycyclic presentation:
2^5,3^2|a^2=e,b^2=d,d^2=e,b^a=bc,d^a=de,f^a=f^2,c^b=ce,f^b=g,g^b=f,f^c=f^2,g^c=g^224 conjugacy classes — view the full character table.
| maximal subgroup of | #10593300, #10593302, #10593304, … |
| maximal normal subgroup of | #10593300, #10593302, #10593304, … |
| maximal quotient of | #10590028, #10590036, #10590038, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 376);