Finite Group DatabasebetaFinite Group Databasebeta| order | 672 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3×C7 · Schur multiplier (H2) = C25 |
| direct product | C3×C7×H(5,𝔽2) |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3×C7 | C24 |
| commutator, Frattini, cosocle | C2 | C24×C3×C7 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C7 ◁ G | |
332 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 168 |
A minimal faithful permutation representation — 4 generators on 18 points:
(1,2,3)(4,5,6,7,8,9,10)(11,12)(13,17)(14,15)(16,18);(12,15)(16,17);(11,12)(13,16)(14,15)(17,18);(11,13)(12,16)(14,18)(15,17)357 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(672, 1252);