Finite Group DatabasebetaFinite Group Databasebeta| order | 448 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
| direct product | Q8×C22⋊D7 |
| Fitting | C22×C7×Q8 ⋊ C2 |
| Sylow | C7 ⋊ D4×Q8 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D7 |
| commutator, socle, cosocle | C22×C7 | C24 |
| Fitting | C22×C7×Q8 | C2 |
| prime radical | C2×C22⋊D7 | C22 |
| hypercenter | C22×Q8 | D7 |
| hypocenter | C7 | D4×Q8 |
| LCS | G ▷ C22×C7 ▷ C7 | |
| UCS | C1 ◁ C22 ◁ C22×Q8 | |
280 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 280 conjugacy classes of subgroups — too many to draw. Only its 123 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 4 generators on 19 points:
(1,2,4,6,7,5,3)(9,11);(2,3)(4,5)(6,7)(8,9)(10,11);(12,14,15,19)(13,16,17,18);(12,13,15,17)(14,18,19,16)85 conjugacy classes — view the full character table.
| maximal subgroup of | #11736250, #11736251, #11736255, … |
| maximal normal subgroup of | #11736250, #11736251, #11736255, … |
| maximal quotient of | #11739607, #11739616, #11739618, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(448, 1268);