Finite Group DatabasebetaFinite Group Databasebeta| order | 672 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C7 · Schur multiplier (H2) = C22 |
| direct product | C3×C7×#137 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3×C7 | C2×D4 |
| commutator, Frattini, cosocle | C4 | C23×C3×C7 |
| prime radical | C3×C7×#41 | C2 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C7 ◁ C2×C3×C4×C7 ◁ G | |
120 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 120 conjugacy classes of subgroups — too many to draw. Only its 80 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 336 |
A minimal faithful permutation representation — 3 generators on 26 points:
(1,2,3)(4,5,6,7,8,9,10)(11,12)(13,19)(14,22)(15,17)(16,20)(18,24)(21,26)(23,25);(11,13,15,21)(12,16,17,25)(14,23,24,20)(18,26,22,19);(12,17)(16,25)(18,22)(19,26)231 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(672, 1042);