Finite Group DatabasebetaFinite Group Databasebeta| order | 648 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Sylow | #731 ⋊ Q8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #4176⋊C2 |
| commutator, Fitting, cosocle | C2×#731 | C22 |
| Frattini | C2×C32 | D32 |
| socle | C2×C3 | ((C3⋊D3)⋊C3)⋊C2 |
| prime radical | C2×H(3,𝔽3) | C2×D3 |
| hypocenter | #731 | Q8 |
| LCS | G ▷ C2×#731 ▷ #731 | |
| UCS | C1 ◁ C2 | |
66 conjugacy classes of subgroups.
| transitive degree | μt(G) = 216 |
Polycyclic presentation:
2^3,3^4|a^2=c,b^2=c,d^3=g^2,b^a=bc,e^a=e^2,f^a=f^2g,d^b=d^2g,f^b=f^2,g^b=g^2,e^d=ef,f^e=fg31 conjugacy classes — view the full character table.
| maximal subgroup of | #4176⋊Q8, #4184⋊Q8, #4177⋊Q8, … |
| maximal normal subgroup of | #4176⋊Q8, #4184⋊Q8, #4177⋊Q8, … |
| maximal quotient of | #13046491, #13046492, #13046493, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(648, 78);