Finite Group DatabasebetaFinite Group Databasebeta| order | 448 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C7 · Schur multiplier (H2) = C25 |
| direct product | C7×#518 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | C24 |
| commutator | C2 | C23×C4×C7 |
| Frattini, cosocle | C22 | C24×C7 |
| prime radical | C22×C7×D4 | C2 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ G | |
338 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 112 |
A minimal faithful permutation representation — 4 generators on 19 points:
(1,2,3,4)(5,6)(7,11)(8,10)(9,12)(13,14,15,16,17,18,19);(5,7)(6,9)(8,12)(10,11);(7,12)(9,11);(6,10)(9,11)238 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(448, 1301);