Finite Group DatabasebetaFinite Group Databasebeta| order | 784 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Sylow | C72 ⋊ M16 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C4 | D72 |
| commutator, socle | C2×C72 | C2×C4 |
| Fitting, cosocle | C4×C72 | C22 |
| prime radical | C2×C7×D7 | C4 |
| hypocenter | C72 | M16 |
| LCS | G ▷ C2×C72 ▷ C72 | |
| UCS | C1 ◁ C4 | |
52 conjugacy classes of subgroups.
| transitive degree | μt(G) = 112 |
Polycyclic presentation:
2^4,7^2|b^2=c,c^2=d,b^a=bd,f^a=f^6,e^b=e^694 conjugacy classes — view the full character table.
| maximal subgroup of | D7×#1131, #13096229, #13096241, … |
| maximal normal subgroup of | D7×#1131, #13096229, #13096241, … |
| maximal quotient of | #13096046, (C2×D7)⋊(C7⋊C8), Dic7⋊(C7⋊C8), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(784, 50);