Finite Group DatabasebetaFinite Group Databasebeta| order | 672 |
| isoclinism | isoclinic class of #10620790 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C4 |
No aliases for this group.
| Fitting, derived | C42×C7 ⋊ C2×C3 |
| Hall | #6437 ⋊ C3 |
| Sylow | C7 ⋊ #127⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, cosocle, hypocenter | C42×C7 | C2×C3 |
| Frattini | C22 | A4×D7 |
| socle | C22×C7 | C2×A4 |
| LCS | G ▷ C42×C7 | |
| UCS | C1 | |
56 conjugacy classes of subgroups.
| transitive degree | μt(G) = 84 |
Polycyclic presentation:
2,3,2^2,7,2^2|c^2=g,d^2=f,c^a=cg,d^a=df,e^a=e^6,c^b=cd,d^b=cf,f^b=g,g^b=fg34 conjugacy classes — view the full character table.
| maximal subgroup of | #857⋊D7, C2×#10620790, (#126⋊C3)⋊D7, … |
| maximal normal subgroup of | #857⋊D7, C2×#10620790, (#126⋊C3)⋊D7, … |
| maximal quotient of | #13052149, #13052152, #13052155, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(672, 372);