Finite Group DatabasebetaFinite Group Databasebeta| order | 448 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C23 |
| direct product | C7×#419 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | C2×D4 |
| commutator | C4 | C22×C4×C7 |
| Frattini, cosocle | C2×C4 | C23×C7 |
| prime radical | C23×C7 | D4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C22×C4×C7 ◁ G | |
150 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2,3,4)(5,6)(7,14)(8,11)(9,13)(10,15)(12,18)(16,19)(17,20)(21,22,23,24,25,26,27);(5,7,9,16)(6,10,13,20)(8,17,18,15)(11,14,12,19);(5,8,9,18)(6,11,13,12)(7,15,16,17)(10,19,20,14)154 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(448, 827);