Finite Group DatabasebetaFinite Group Databasebeta| order | 1008 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22×C3 |
No aliases for this group.
| Hall | C32×C7 ⋊ #41 |
| Sylow | C32 ⋊ #1157 |
| Sylow | C7 ⋊ #3830 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×D7×C3⋊D3 |
| commutator, socle, cosocle | C2×C32×C7 | C23 |
| Fitting | C32×C4×C7 | C22 |
| hypercenter | C4 | D7×C3⋊D3 |
| hypocenter | C32×C7 | #41 |
| LCS | G ▷ C2×C32×C7 ▷ C32×C7 | |
| UCS | C1 ◁ C2 ◁ C4 | |
240 conjugacy classes of subgroups.
| transitive degree | μt(G) = 504 |
Polycyclic presentation:
2^4,3^2,7|b^2=d,c^2=d,c^a=cd,g^a=g^6,e^b=e^2,f^b=f^2102 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1008, 727);