Finite Group DatabasebetaFinite Group Databasebeta| order | 448 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C7 · Schur multiplier (H2) = C24×C4 |
| direct product | C2×C7×#124 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C7 | C23 |
| commutator, Frattini, cosocle | C22 | C24×C7 |
| prime radical | C22×C7×D4 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×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);(4,5)(14,15)(16,19)(17,18)(20,21);(3,4,6,5)(15,18)(19,21);(1,2)(3,5,6,4)(14,16,17,20)(15,19,18,21)196 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(448, 1309);