Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| isoclinism | isoclinic class of C7⋊#4625 |
| automorphism | ? |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | C7⋊#4625 |
| derived | C7×C22⋊A4 ⋊ C4 |
| Sylow | C7 ⋊ #4625 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, hypocenter | C7×C22⋊A4 | C4 |
| Frattini | C23 | C7⋊S4 |
| Fitting | C7×#120 | D3 |
| socle | C22×C7 | SL(2,ℤ/4) |
| prime radical, cosocle | C7×#120⋊C3 | C2 |
| LCS | G ▷ C7×C22⋊A4 | |
| UCS | C1 | |
76 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 112 |
Polycyclic presentation:
2^2,3,7,2^4|a^2=b,c^a=c^2,d^a=d^6,e^a=efg,f^a=fgh,g^a=gh,e^b=eg,f^b=fh,e^c=f,f^c=ef,g^c=h,h^c=gh41 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 831);