Finite Group DatabasebetaFinite Group Databasebeta| order | 648 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2×C3 |
No aliases for this group.
| Sylow | C34 ⋊ D4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | C3⋊#1113 |
| commutator, Fitting, socle, cosocle | C2×C34 | C22 |
| hypocenter | C34 | D4 |
| LCS | G ▷ C2×C34 ▷ C34 | |
| UCS | C1 ◁ C2 | |
426 conjugacy classes of subgroups.
| transitive degree | μt(G) = 72 |
Polycyclic presentation:
2^3,3^4|b^a=bc,d^a=d^2,e^a=e^2,g^a=g^2,d^b=d^2,e^b=e^2,f^b=f^263 conjugacy classes — view the full character table.
| maximal subgroup of | #13049434, #13049435, #13049446, … |
| maximal normal subgroup of | #13049434, #13049435, #13049446, … |
| maximal quotient of | #13049086, #13049088, #13049091, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(648, 675);