Finite Group DatabasebetaFinite Group Databasebeta| order | 448 |
| isoclinism | isoclinic class of D4×AGL(1,𝔽8) |
| automorphism | Aut(G) = D4×AΓL(1,𝔽8) (order 1344) · Inn(G) = C22×AGL(1,𝔽8) · Out(G) = C2×C3 |
| homology | Abelianization (H1) = C22×C7 · Schur multiplier (H2) = C2 |
| Fitting, Sylow | C23×D4 ⋊ C7 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C22×AGL(1,𝔽8) |
| commutator, socle, cosocle | C24 | C22×C7 |
| Fitting | C23×D4 | C7 |
| hypercenter | D4 | AGL(1,𝔽8) |
| hypocenter | C23 | C7×D4 |
| LCS | G ▷ C24 ▷ C23 | |
| UCS | C1 ◁ C2 ◁ D4 | |
127 conjugacy classes of subgroups.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 28 |
A minimal faithful permutation representation — 2 generators on 12 points:
(2,4)(6,7,8,10,12,9,11);(1,2)(3,4)(5,9)(6,7)(8,12)(10,11)40 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(448, 1363);