Finite Group DatabasebetaFinite Group Databasebeta| order | 448 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C7 · Schur multiplier (H2) = C29 |
| direct product | C22×C7×#41 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C7 | C22 |
| commutator, Frattini, cosocle | C2 | C25×C7 |
| socle | C23×C7 | C23 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C7 ◁ G | |
890 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 5 generators on 19 points:
(1,2)(5,6,7,8,9,10,11);(3,4);(13,16)(17,19);(12,13)(14,17)(15,16)(18,19);(1,2)(3,4)(12,14,15,18)(13,17,16,19)280 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(448, 1388);