Finite Group DatabasebetaFinite Group Databasebeta| order | 672 |
| isoclinism | isoclinic class of #10621175 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Fitting, Hall | C7×#126 ⋊ C3 |
| derived | C42×C7 ⋊ C2×C3 |
| Sylow | #126 ⋊ ΓL(1,𝔽8) |
| Sylow | C7 ⋊ #126⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C42×C7 | C2×C3 |
| Frattini | C22 | C2×C22⋊ΓL(1,𝔽8) |
| Fitting | C7×#126 | C3 |
| socle | C22×C7 | C2×A4 |
| LCS | G ▷ C42×C7 | |
| UCS | C1 | |
40 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 112 |
Polycyclic presentation:
2,3,2^2,7,2^2|c^2=f,d^2=g,c^a=cg,d^a=dfg,c^b=df,d^b=cd,e^b=e^2,f^b=g,g^b=fg38 conjugacy classes — view the full character table.
| maximal subgroup of | #13057647, #13057649, C2×#10621175, … |
| maximal normal subgroup of | #13057647, #13057649, C2×#10621175, … |
| maximal quotient of | #13055064, #13055069, #13055072, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(672, 757);