Finite Group DatabasebetaFinite Group Databasebeta| order | 448 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of D32 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C7 · Schur multiplier (H2) = C1 |
| direct product | C7×SD64 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C7 | D16 |
| commutator, Frattini, cosocle | C16 | C22×C7 |
| prime radical | C7×D16 | C2 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7 ◁ C4×C7 ◁ C7×C8 ◁ C7×C16 ◁ G | |
32 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 2 generators on 39 points:
(1,2,3,4,5,6,7)(8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39);(9,23)(10,38)(11,21)(12,36)(13,19)(14,34)(15,17)(16,32)(18,30)(20,28)(22,26)(25,39)(27,37)(29,35)(31,33)133 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(448, 176);