Finite Group DatabasebetaFinite Group Databasebeta| order | 648 |
| automorphism | ? |
| homology | Abelianization (H1) = C33 · Schur multiplier (H2) = C32 |
No aliases for this group.
| Sylow | Q8 ⋊ #737 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle | C2×C3 | C32×A4 |
| commutator, cosocle | C3×Q8 | C33 |
| Fitting | Q8×H(3,𝔽3) | C3 |
| prime radical, hypercenter | C2×H(3,𝔽3) | A4 |
| hypocenter | Q8 | #737 |
| LCS | G ▷ C3×Q8 ▷ Q8 | |
| UCS | C1 ◁ C2×C3 ◁ C2×H(3,𝔽3) | |
110 conjugacy classes of subgroups.
| transitive degree | μt(G) = 216 |
Polycyclic presentation:
3^4,2^3|a^3=d,e^2=g,f^2=g,e^a=f,f^a=efg,c^b=cd,f^e=fg77 conjugacy classes — view the full character table.
| maximal subgroup of | #13048386, #13048387, #13048462, … |
| maximal normal subgroup of | #13048386, #13048387, #13048462, … |
| maximal quotient of | C2×#10619913, #14517173, #14517174, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(648, 432);