Finite Group DatabasebetaFinite Group Databasebeta| order | 1512 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Hall | C7×H(3,𝔽3) ⋊ D4 |
| Sylow | H(3,𝔽3) ⋊ C22⋊D7 |
| Sylow | C7 ⋊ C4⋊#272 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2×C3 | D7×C3⋊D3 |
| commutator, Fitting, cosocle | C2×C7×H(3,𝔽3) | C22 |
| socle | C2×C3×C7 | C2×C3⋊D3 |
| hypocenter | C7×H(3,𝔽3) | D4 |
| LCS | G ▷ C2×C7×H(3,𝔽3) ▷ C7×H(3,𝔽3) | |
| UCS | C1 ◁ C2×C3 | |
176 conjugacy classes of subgroups.
| transitive degree | μt(G) = 252 |
Polycyclic presentation:
2^3,3^2,7,3|a^2=c,b^a=bc,f^a=f^6,d^b=d^2,e^b=e^2,e^d=eg91 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1512, 263);