Finite Group DatabasebetaFinite Group Databasebeta| order | 648 |
| isoclinism | isoclinic class of #10620185 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Fitting, socle | C33 ⋊ S4 |
| derived | #1113⋊C3 ⋊ C2 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | #1113⋊C3 | C2 |
| Fitting, socle | C33 | S4 |
| LCS | G ▷ #1113⋊C3 | |
| UCS | C1 | |
58 conjugacy classes of subgroups.
| minimal degree | μ(G) = 9 |
| transitive degree | μt(G) = 9 |
Polycyclic presentation:
2,3,2^2,3^3|b^a=b^2,c^a=cd,e^a=f,f^a=e,c^b=d,d^b=cd,e^b=f,f^b=g,g^b=e,e^c=e^2,g^c=g^2,e^d=e^2,f^d=f^217 conjugacy classes — view the full character table.
| maximal subgroup of | #10620184⋊C2, C2×#10620185, C3×#10620185, … |
| maximal normal subgroup of | #10620184⋊C2, C2×#10620185, C3×#10620185 |
| maximal quotient of | #13049366, #13049367, #13049368, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(648, 704);