Finite Group DatabasebetaFinite Group Databasebeta| order | 784 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C72 ⋊ D8 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #87640 |
| commutator, Fitting, cosocle | C4×C72 | C22 |
| Frattini, hypercenter | C4 | D72 |
| socle | C2×C72 | D4 |
| hypocenter | C72 | D8 |
| LCS | G ▷ C4×C72 ▷ C2×C72 ▷ C72 | |
| UCS | C1 ◁ C2 ◁ C4 | |
56 conjugacy classes of subgroups.
| transitive degree | μt(G) = 112 |
Polycyclic presentation:
2^4,7^2|c^2=d,b^a=bc,c^a=cd,f^a=f^6,c^b=cd,e^b=e^673 conjugacy classes — view the full character table.
| maximal subgroup of | #13096187, #13096188, #13096235, … |
| maximal normal subgroup of | #13096187, #13096188, #13096235, … |
| maximal quotient of | #13096037, #13096039, #13096042, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(784, 52);