Finite Group DatabasebetaFinite Group Databasebeta| order | 672 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C7 · Schur multiplier (H2) = C23 |
| direct product | C2×C3×C7×D8 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3×C7 | D4 |
| commutator, Frattini, cosocle | C4 | C23×C3×C7 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3×C7 ◁ C2×C3×C4×C7 ◁ G | |
152 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 336 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2)(11,12,13)(14,15,16,17,18,19,20);(3,4,5,6,7,8,9,10);(4,10)(5,9)(6,8)294 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(672, 1037);