Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| automorphism | Aut(G) = C2×D4×GO+(4,𝔽2) (order 1152) · Inn(G) = C2×GO+(4,𝔽2) · Out(G) = C23 |
| 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)(7,10,12,11)(8,14,15,20)(9,17,19,18)(13,16,21,22);(3,4)(7,9,12,19)(8,13,15,21)(10,18,11,17)(14,22,20,16);(1,2)(3,6)(4,5)(7,8)(9,16)(10,14)(11,20)(12,15)(13,18)(17,21)(19,22)Polycyclic presentation:
2^5,3^2|b^2=e,c^2=e,d^2=e,b^a=bd,d^a=de,f^a=f^2,c^b=ce,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 | #10593300, #10593302, #10593304, … |
| maximal normal subgroup of | #10593300, #10593302, #10593304, … |
| maximal quotient of | #10593347, #10593358, #10593360, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 874);