Finite Group DatabasebetaFinite Group Databasebeta| order | 648 |
| isoclinism | isoclinic class of #10620204 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Fitting, socle, Sylow | C34 ⋊ D4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, cosocle | C32×C3⋊D3 | C22 |
| Fitting, socle, hypocenter | C34 | D4 |
| LCS | G ▷ C32×C3⋊D3 ▷ C34 | |
| UCS | C1 | |
187 conjugacy classes of subgroups; only the 18 normal subgroups are shown.
This group has 187 conjugacy classes of subgroups — too many to draw. Only its 18 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 12 |
Polycyclic presentation:
2^3,3^4|b^2=c,b^a=bc,d^a=d^2,g^a=g^2,d^b=e,e^b=d^2,f^b=f^2,d^c=d^2,e^c=e^230 conjugacy classes — view the full character table.
| maximal subgroup of | D3×D32⋊D3, (D32⋊D3)⋊D3, D3×#157⋊D3, … |
| maximal normal subgroup of | D3×D32⋊D3, (D32⋊D3)⋊D3, D3×#157⋊D3, … |
| maximal quotient of | #13049443, #13049444, #13049445, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(648, 723);