Finite Group DatabasebetaFinite Group Databasebeta| order | 672 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4×C7 · Schur multiplier (H2) = C2×C4 |
| direct product | C3×C7×#117 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4×C7 | C22 |
| commutator | C2 | C22×C3×C4×C7 |
| Frattini, cosocle | C22 | C23×C3×C7 |
| socle | C22×C3×C7 | C23 |
| prime radical | C23×C3×C7 | C22 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4×C7 ◁ G | |
152 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 336 |
A minimal faithful permutation representation — 3 generators on 22 points:
(1,2,3)(4,5,6,7,8,9,10)(11,12,13,14)(15,16)(17,20)(18,19)(21,22);(16,19)(20,22);(15,17,18,21)(16,20,19,22)420 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(672, 1022);