Finite Group DatabasebetaFinite Group Databasebeta| order | 672 |
| isoclinism | isoclinic class of #10620804 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting, derived | C24×C7 ⋊ C2×C3 |
| Hall | #6516 ⋊ C3 |
| Sylow | C7 ⋊ #120⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, cosocle, hypocenter | C24×C7 | C2×C3 |
| Frattini | C22 | D7⋊A4 |
| socle | C22×C7 | C2×A4 |
| LCS | G ▷ C24×C7 | |
| UCS | C1 | |
70 conjugacy classes of subgroups.
| transitive degree | μt(G) = 56 |
Polycyclic presentation:
2,3,2^4,7|c^a=cef,d^a=df,g^a=g^6,c^b=cd,d^b=c,e^b=f,f^b=ef,g^b=g^426 conjugacy classes — view the full character table.
| maximal subgroup of | D7⋊(#120⋊C3), #13057645, C2×#10620804, … |
| maximal normal subgroup of | D7⋊(#120⋊C3), #13057645, C2×#10620804, … |
| maximal quotient of | #13052180, #13052181, #13052182, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(672, 386);