Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| automorphism | Aut(G) = C22×D32×D4 (order 1152) · Inn(G) = C2×D32 · Out(G) = C24 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C32 ⋊ #131 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C4 | C2×D32 |
| commutator, socle | C2×C32 | C22×C4 |
| Fitting | C2×C32×C4 | C22 |
| prime radical | C3×#235 | C2 |
| cosocle | C32×C4 | C23 |
| hypercenter | C2×C4 | D32 |
| hypocenter | C32 | #131 |
| LCS | G ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C4 ◁ C2×C4 | |
135 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 135 conjugacy classes of subgroups — too many to draw. Only its 60 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:
(2,7)(3,9)(8,14)(10,15)(17,18,19)(20,21,22);(1,3,4,10,5,9,11,15)(2,6,8,12,7,13,14,16)(18,19);(1,2)(3,6)(4,8)(5,7)(9,13)(10,12)(11,14)(15,16)(21,22)Polycyclic presentation:
2^5,3^2|b^2=d,d^2=e,c^a=ce,g^a=g^2,c^b=ce,f^b=f^248 conjugacy classes — view the full character table.
| maximal subgroup of | #10590389, #10590390, #10590919, … |
| maximal normal subgroup of | #10590389, #10590390, #10590919, … |
| maximal quotient of | #10590250, #10590251, #10590253, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 463);