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) ⋊ D28 |
| Sylow | C7 ⋊ #6198 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #272⋊D7 |
| commutator, Fitting, cosocle | C2×C7×H(3,𝔽3) | C22 |
| Frattini | C2×C3 | (C3⋊D3)⋊D7 |
| socle | C2×C3×C7 | D32 |
| hypocenter | C7×H(3,𝔽3) | D4 |
| LCS | G ▷ C2×C7×H(3,𝔽3) ▷ C7×H(3,𝔽3) | |
| UCS | C1 ◁ C2 | |
124 conjugacy classes of subgroups.
| transitive degree | μt(G) = 252 |
Polycyclic presentation:
2^3,3^2,7,3|b^2=c,b^a=bc,d^a=d^2,f^a=f^6,g^a=g^2,d^b=d^2,e^b=e^2,e^d=eg79 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1512, 307);