Finite Group DatabasebetaFinite Group Databasebeta| order | 1512 |
| isoclinism | isoclinic class of #272⋊D7 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| derived, Hall | C7×H(3,𝔽3) ⋊ C2×C4 |
| Sylow | H(3,𝔽3) ⋊ C4×D7 |
| Sylow | C7 ⋊ ((C3⋊D3)⋊C3)⋊C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #272⋊D7 |
| commutator, hypocenter | C7×H(3,𝔽3) | C2×C4 |
| Frattini | C2×C3 | (C3⋊D3)⋊D7 |
| Fitting, cosocle | C2×C7×H(3,𝔽3) | C22 |
| socle | C2×C3×C7 | D32 |
| prime radical | C2×#67385 | C2 |
| LCS | G ▷ C7×H(3,𝔽3) | |
| UCS | C1 ◁ C2 | |
124 conjugacy classes of subgroups; only the 31 normal subgroups are shown.
This group has 124 conjugacy classes of subgroups — too many to draw. Only its 31 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 252 |
Polycyclic presentation:
2^3,3^2,7,3|b^2=c,d^a=d^2,f^a=f^6,g^a=g^2,d^b=d^2,e^b=e^2,e^d=eg82 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1512, 306);