Finite Group DatabasebetaFinite Group Databasebeta| order | 672 |
| isoclinism | isoclinic class of ΓL(1,𝔽8) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C16 · Schur multiplier (H2) = C2 |
| direct product | C2×C16×ΓL(1,𝔽8) |
| unambiguous semidirect product | (C2×C7×C16)⋊C3 |
| Fitting, Hall | C2×C7×C16 ⋊ C3 |
| derived, Sylow | C7 ⋊ C2×C3×C16 |
56 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 2 generators on 25 points:
(1,2)(20,21,22)(23,25,24);(3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18)(19,20,21,23,22,24,25)160 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(672, 129);