Finite Group DatabasebetaFinite Group Databasebeta| order | 448 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C7 · Schur multiplier (H2) = C24×C4 |
| direct product | C2×C7×#117 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C7 | C22 |
| commutator | C2 | C23×C4×C7 |
| Frattini, cosocle | C22 | C24×C7 |
| socle | C23×C7 | C23 |
| prime radical | C24×C7 | C22 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C7 ◁ G | |
330 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 4 generators on 21 points:
(1,2)(7,8,9,10,11,12,13);(3,4,5,6)(15,18)(19,21);(14,15)(16,19)(17,18)(20,21);(1,2)(14,16,17,20)(15,19,18,21)280 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(448, 1297);