Finite Group DatabasebetaFinite Group Databasebeta| order | 648 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C3×H(3,𝔽3) ⋊ D4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #66432 |
| commutator, Fitting, cosocle | C2×C3×H(3,𝔽3) | C22 |
| Frattini | C2×C3 | #1113 |
| socle | C2×C32 | D32 |
| hypocenter | C3×H(3,𝔽3) | D4 |
| LCS | G ▷ C2×C3×H(3,𝔽3) ▷ C3×H(3,𝔽3) | |
| UCS | C1 ◁ C2 | |
214 conjugacy classes of subgroups; only the 31 normal subgroups are shown.
This group has 214 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) = 108 |
Polycyclic presentation:
2^3,3^4|a^2=c,b^a=bc,d^a=d^2,f^a=f^2,d^b=d^2,e^b=e^2,g^b=g^2,f^d=fg39 conjugacy classes — view the full character table.
| maximal subgroup of | #13048253, #13048255, #13048265, … |
| maximal normal subgroup of | #13048253, #13048255, #13048265, … |
| maximal quotient of | #13047744, #13047745, #13047746, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(648, 421);