Finite Group DatabasebetaFinite Group Databasebeta| order | 672 |
| isoclinism | isoclinic class of #4245 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
| direct product | C2×Dic7⋊Dic3 |
| Hall | C3×C7 ⋊ C2×C4⋊C4 |
| Sylow | C7 ⋊ C2×C4⋊Dic3 |
| Sylow | C3 ⋊ C2×Dic7⋊C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23 | D3×D7 |
| commutator | C2×C3×C7 | C22×C4 |
| Frattini | C22 | C2×D3×D7 |
| Fitting, socle, prime radical | C23×C3×C7 | C22 |
| cosocle | C22×C3×C7 | C23 |
| hypocenter | C3×C7 | C2×C4⋊C4 |
| LCS | G ▷ C2×C3×C7 ▷ C3×C7 | |
| UCS | C1 ◁ C23 | |
184 conjugacy classes of subgroups.
| transitive degree | μt(G) = 672 |
108 conjugacy classes — view the full character table.
| maximal subgroup of | Dic3×Dic7⋊C4, #13053880, #13053882, … |
| maximal normal subgroup of | Dic3×Dic7⋊C4, #13053880, #13053882, … |
| maximal quotient of | #13053696, C2×Dic7⋊(C3⋊C8), #13053826, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(672, 685);