Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| isoclinism | isoclinic class of #3714 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
| direct product | C2×#3714 |
| Sylow | C32 ⋊ C2×D8 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | #626 |
| commutator, cosocle | C32×C4 | C23 |
| Frattini | C4 | C2×D32 |
| Fitting | C2×C32×C4 | C22 |
| socle | C22×C32 | D4 |
| hypercenter | C2×C4 | D32 |
| hypocenter | C32 | C2×D8 |
| LCS | G ▷ C32×C4 ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C22 ◁ C2×C4 | |
179 conjugacy classes of subgroups; only the 52 normal subgroups are shown.
This group has 179 conjugacy classes of subgroups — too many to draw. Only its 52 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 3 generators on 16 points:
(1,2)(4,5)(6,7,8)(9,10,14,16,12,15,11,13);(3,4,5)(7,8)(10,13)(11,14)(15,16);(1,2)48 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#3714, #10590346, #10590349, … |
| maximal normal subgroup of | C4×#3714, #10590346, #10590349, … |
| maximal quotient of | #10590272, #10590273, #10590281, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 472);