Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| isoclinism | isoclinic class of #63734 |
| automorphism | Aut(G) = #11860179 (order 1152) · Inn(G) = #3774 · Out(G) = C2×C4 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Sylow | C32 ⋊ SD32 |
46 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)(7,8,11,14)(9,16,18,22)(10,19,17,13)(12,20,21,15);(1,2)(3,6)(4,5)(8,12)(9,15)(10,17)(13,22)(14,21)(16,19)(18,20)Polycyclic presentation:
2^5,3^2|b^2=e,c^2=de,d^2=e,b^a=bc,c^a=cd,d^a=de,f^a=f^2,c^b=cd,d^b=de,f^b=g,g^b=f,f^c=f^2,g^c=g^221 conjugacy classes — view the full character table.
| maximal subgroup of | #10593275, #10593277, #10593278, … |
| maximal normal subgroup of | #10593275, #10593277, #10593278, … |
| maximal quotient of | #10590061, #10590062, #10590069, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 383);