Finite Group DatabasebetaFinite Group Databasebeta| order | 448 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C7 · Schur multiplier (H2) = C25 |
| direct product | C2×C7×#95 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24×C7 | C22 |
| commutator | C2 | C2×C42×C7 |
| Frattini, cosocle | C23 | C23×C7 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24×C7 ◁ G | |
330 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 3 generators on 21 points:
(1,2)(15,16,17,18,19,20,21);(8,10)(11,12,13,14);(3,4,5,6)(7,8)(9,10)280 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(448, 783);