Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| isoclinism | isoclinic class of GO+(4,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
| direct product | C2×D32⋊C4 |
| Sylow | C32 ⋊ C2×C22⋊C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | GO+(4,𝔽2) |
| commutator | C3⋊D3 | C22×C4 |
| Frattini | C2 | C2×GO+(4,𝔽2) |
| Fitting, socle | C22×C32 | D4 |
| prime radical | C22×D32 | C2 |
| cosocle | C2×C3⋊D3 | C23 |
| hypocenter | C32 | C2×C22⋊C4 |
| LCS | G ▷ C3⋊D3 ▷ C32 | |
| UCS | C1 ◁ C22 | |
226 conjugacy classes of subgroups; only the 43 normal subgroups are shown.
This group has 226 conjugacy classes of subgroups — too many to draw. Only its 43 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 3 generators on 12 points:
(3,4,5,6)(7,10,9)(8,12);(7,8)(9,11)(10,12);(1,2)36 conjugacy classes — view the full character table.
| maximal subgroup of | C4×D32⋊C4, #10593398, #10593399, … |
| maximal normal subgroup of | C4×D32⋊C4, #10593398, #10593399, … |
| maximal quotient of | C2×D32⋊C8, #10593307, #10593308, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 880);