Finite Group DatabasebetaFinite Group Databasebeta| order | 672 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C7 · Schur multiplier (H2) = C2 |
| direct product | C7×#986 |
| Fitting | C7×#41 ⋊ D3 |
| derived | SL(2,𝔽3) ⋊ C22×C7 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C7 | C2×S4 |
| commutator, cosocle, hypocenter | SL(2,𝔽3) | C22×C7 |
| Frattini | C2 | C2×C7×S4 |
| Fitting | C7×#41 | D3 |
| hypercenter | C4×C7 | S4 |
| LCS | G ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C2×C7 ◁ C4×C7 | |
84 conjugacy classes of subgroups.
| transitive degree | μt(G) = 112 |
91 conjugacy classes — view the full character table.
| maximal subgroup of | #13057082, #13057084, #13057088, … |
| maximal normal subgroup of | #13057082, #13057084, #13057088, … |
| maximal quotient of | C7×#5391, C7×#5394, C7×C4⋊GL(2,𝔽3), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(672, 1075);