Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| isoclinism | isoclinic class of #625 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C22×#626 |
| Sylow | C32 ⋊ C22×D4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23 | D32 |
| commutator, cosocle | C2×C32 | C24 |
| Frattini | C2 | C22×D32 |
| Fitting, socle | C23×C32 | C22 |
| hypocenter | C32 | C22×D4 |
| LCS | G ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C23 | |
539 conjugacy classes of subgroups; only the 148 normal subgroups are shown.
This group has 539 conjugacy classes of subgroups — too many to draw. Only its 148 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 4 generators on 14 points:
(1,2)(3,4)(6,7)(8,9,10)(11,12,13,14);(5,6,7)(9,10)(12,14);(3,4);(1,2)60 conjugacy classes — view the full character table.
| maximal subgroup of | #10591275, #10591291, #10591292, … |
| maximal normal subgroup of | #10591275, #10591291, #10591292, … |
| maximal quotient of | C22×#3714, C2×#63825, C22×#3716, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 974);