Finite Group DatabasebetaFinite Group Databasebeta| order | 784 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Sylow | C72 ⋊ SD16 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | #5997⋊C2 |
| commutator, cosocle | C7⋊Dic7 | C22 |
| Fitting, socle | C2×C72 | D4 |
| prime radical | #87640 | C2 |
| hypocenter | C72 | SD16 |
| LCS | G ▷ C7⋊Dic7 ▷ C2×C72 ▷ C72 | |
| UCS | C1 ◁ C2 | |
37 conjugacy classes of subgroups.
| transitive degree | μt(G) = 56 |
Polycyclic presentation:
2^4,7^2|b^2=d,c^2=d,b^a=bc,c^a=cd,e^a=e^6,c^b=cd,e^b=f,f^b=e,e^c=e^6,f^c=f^637 conjugacy classes — view the full character table.
| maximal subgroup of | #13096637, #13096638, #13096640, … |
| maximal normal subgroup of | #13096637, #13096638, #13096640, … |
| maximal quotient of | #13096192, #13096193, #13096195, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(784, 108);