Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| isoclinism | isoclinic class of #3714 |
| automorphism | Aut(G) = C22×D32×D4 (order 1152) · Inn(G) = #626 · Out(G) = C24 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Sylow | C32 ⋊ #108 |
70 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 2 generators on 22 points:
(1,3,4,12,6,13,5,14)(2,7,8,11,10,15,9,16)(18,19)(20,21,22);(1,2,5,9,6,10,4,8)(3,11,12,15,13,16,14,7)(17,18,19)(21,22)Polycyclic presentation:
2^5,3^2|a^2=d,b^2=c,c^2=e,d^2=e,b^a=bd,g^a=g^2,d^b=de,f^b=f^248 conjugacy classes — view the full character table.
| maximal subgroup of | #10590341, #10590342, D3×#819, … |
| maximal normal subgroup of | #10590341, #10590342, D3×#819, … |
| maximal quotient of | #10589097, #10589099, #10589166, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 225);