Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| automorphism | Aut(G) = C22×D32×D4 (order 1152) · Inn(G) = #3802 · Out(G) = C23 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C32 ⋊ #137 |
130 conjugacy classes of subgroups; only the 38 normal subgroups are shown.
This group has 130 conjugacy classes of subgroups — too many to draw. Only its 38 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 3 generators on 22 points:
(1,3,4,9,5,11,12,16)(2,6,7,14,8,15,13,10)(17,18,19)(20,21,22);(1,2,5,8)(3,10,11,14)(4,13,12,7)(6,9,15,16)(18,19);(3,9)(4,12)(6,14)(7,13)(10,15)(11,16)(21,22)Polycyclic presentation:
2^5,3^2|b^2=e,c^2=d,d^2=e,c^a=cd,d^a=de,g^a=g^2,c^b=cde,d^b=de,f^b=f^233 conjugacy classes — view the full character table.
| maximal subgroup of | #10594685, #10594686, C2×#63811, … |
| maximal normal subgroup of | #10594685, #10594686, C2×#63811, … |
| maximal quotient of | #10590650, #10590686, #10590688, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 459);