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 | C2×C7×#103 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C7 | D4 |
| commutator | C4 | C22×C4×C7 |
| Frattini, cosocle | C2×C4 | C23×C7 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C7 ◁ C22×C4×C7 ◁ G | |
162 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 3 generators on 21 points:
(1,2)(7,8,9,10,11,12,13);(3,4,5,6)(15,18)(16,20)(19,21);(3,4)(5,6)(14,15,17,19)(16,21,20,18)196 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(448, 823);