Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| isoclinism | isoclinic class of #3774 |
| automorphism | Aut(G) = C2×#10596417 (order 1152) · Inn(G) = GO+(4,𝔽2) · Out(G) = C2×D4 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Sylow | C32 ⋊ #108 |
50 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)(7,9,11,17,12,18,10,20)(8,13,14,21,15,22,19,16);(1,2)(3,6)(4,5)(7,8,11,14,12,15,10,19)(9,16,17,13,18,21,20,22)Polycyclic presentation:
2^5,3^2|a^2=d,b^2=d,c^2=e,d^2=e,b^a=bc,c^a=ce,f^a=f^2,c^b=ce,f^b=g,g^b=f,f^c=f^2,g^c=g^230 conjugacy classes — view the full character table.
| maximal subgroup of | #10593163, #10593297, #10593298, … |
| maximal normal subgroup of | #10593163, #10593297, #10593298, … |
| maximal quotient of | #10590041, #10590042, #10590051, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 381);