Finite Group DatabasebetaFinite Group Databasebeta| order | 672 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C7 · Schur multiplier (H2) = C1 |
| direct product | C3×C7×SD32 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3×C7 | D8 |
| commutator, Frattini, cosocle | C8 | C22×C3×C7 |
| prime radical | C3×C7×D8 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C7 ◁ C3×C4×C7 ◁ C3×C7×C8 ◁ G | |
52 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 336 |
A minimal faithful permutation representation — 2 generators on 26 points:
(1,2,3)(4,5,6,7,8,9,10)(11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26);(12,18)(13,25)(14,16)(15,23)(17,21)(20,26)(22,24)231 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(672, 276);