Finite Group DatabasebetaFinite Group Databasebeta| order | 672 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C1 |
| direct product | SL(2,𝔽3)×Dic7 |
| derived | C7×Q8 ⋊ C3×C4 |
| Hall | Q8×Dic7 ⋊ C3 |
| Sylow | C7 ⋊ C4×SL(2,𝔽3) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C22 | A4×D7 |
| commutator, hypocenter | C7×Q8 | C3×C4 |
| Fitting, cosocle | C2×C7×Q8 | C2×C3 |
| socle | C22×C7 | C2×A4 |
| prime radical | C2×C7×SL(2,𝔽3) | C2 |
| LCS | G ▷ C7×Q8 | |
| UCS | C1 ◁ C22 | |
60 conjugacy classes of subgroups.
| transitive degree | μt(G) = 224 |
70 conjugacy classes — view the full character table.
| maximal subgroup of | Dic7×#226, #13057052, #13057054, … |
| maximal normal subgroup of | Dic7×#226, #13057052, #13057054, … |
| maximal quotient of | SL(2,𝔽3)×C7⋊C8, C2×SL(2,𝔽3)×Dic7, SL(2,𝔽3)×C7⋊AGL(1,𝔽5) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(672, 376);