Finite Group DatabasebetaFinite Group Databasebeta| order | 448 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C23 |
| direct product | C7×#416 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C7 | D4 |
| commutator | C4 | C22×C4×C7 |
| Frattini, cosocle | C2×C4 | C23×C7 |
| socle | C22×C7 | C2×D4 |
| prime radical | C2×C7×#41 | C2 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C7 ◁ C22×C4×C7 ◁ G | |
158 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)(10,16)(12,18)(15,19)(17,20)(21,22,23,24,25,26,27);(5,7)(6,10)(8,15)(9,17)(11,14)(12,20)(13,19)(16,18);(5,8,9,18)(6,11,13,12)(7,15,17,16)(10,14,19,20)196 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(448, 824);