Finite Group DatabasebetaFinite Group Databasebeta| order | 672 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4×C7 · Schur multiplier (H2) = C2 |
| direct product | C3×C7×#101 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3×C7 | C22⋊C4 |
| commutator | C22 | C2×C3×C4×C7 |
| Frattini, cosocle | C2×C4 | C22×C3×C7 |
| prime radical | C22×C3×C7 | C2×C4 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C7 ◁ C2×C3×C4×C7 ◁ G | |
76 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)(11,12,14,18,15,17,23,25)(13,19,20,26,21,16,22,24);(11,13,15,21)(12,16,17,19)(14,22,23,20)(18,26,25,24)231 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(672, 265);