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 ⋊ #137 |
99 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 3 generators on 22 points:
(1,3,4)(2,5,6)(8,13)(12,21)(14,18)(15,22);(3,4)(7,9,11,17)(8,12,13,21)(10,19,20,16)(14,22,18,15);(1,2)(3,6)(4,5)(7,8)(9,15)(10,18)(11,13)(12,16)(14,20)(17,22)(19,21)Polycyclic presentation:
2^5,3^2|b^2=e,d^2=e,b^a=bd,c^a=ce,d^a=de,f^a=f^2,d^b=de,f^b=g,g^b=f,f^d=f^2,g^d=g^224 conjugacy classes — view the full character table.
| maximal subgroup of | #10593452, #10593529, #10596499, … |
| maximal normal subgroup of | #10593452, #10593529, #10596499, … |
| maximal quotient of | #10593352, #10593362, #10593367, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 887);