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) = C22 |
| direct product | C3×C4×C7×Q8 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4×C7 | C22 |
| commutator | C2 | C22×C3×C4×C7 |
| Frattini, cosocle | C22 | C23×C3×C7 |
| socle, prime radical | C22×C3×C7 | C23 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4×C7 ◁ G | |
140 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 672 |
A minimal faithful permutation representation — 3 generators on 22 points:
(1,2,3)(4,5,6,7)(16,17,18,19,20,21,22);(8,9,10,11)(12,15,14,13);(8,12,10,14)(9,13,11,15)420 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(672, 1024);