Finite Group DatabasebetaFinite Group Databasebeta| order | 648 |
| isoclinism | isoclinic class of #4207⋊C2 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | #4207⋊C4 |
| derived, Sylow | #737 ⋊ C2×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #4207⋊C2 |
| commutator, hypocenter | #737 | C2×C4 |
| Frattini, socle | C2×C3 | #1113 |
| Fitting, cosocle | C2×#737 | C22 |
| prime radical | C2×#4207 | C2 |
| LCS | G ▷ #737 | |
| UCS | C1 ◁ C2 | |
117 conjugacy classes of subgroups; only the 29 normal subgroups are shown.
This group has 117 conjugacy classes of subgroups — too many to draw. Only its 29 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,e^3=g,d^a=d^2,f^a=f^2,d^b=d^2,e^b=e^2g^2,g^b=g^2,f^d=fg^242 conjugacy classes — view the full character table.
| maximal subgroup of | #4207⋊Q8, #13048756, #13048757, … |
| maximal normal subgroup of | #4207⋊Q8, #13048756, #13048757, … |
| maximal quotient of | #4207⋊C8, #13047754, #13047774, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(648, 425);