Finite Group DatabasebetaFinite Group Databasebeta| order | 784 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2×C7 |
No aliases for this group.
| Sylow | C72 ⋊ C4⋊C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C2×C7⋊D7 |
| commutator | C2×C72 | C2×C4 |
| Fitting | C2×C4×C72 | C2 |
| socle, prime radical, cosocle | C22×C72 | C22 |
| hypercenter | C2×C4 | C7⋊D7 |
| hypocenter | C72 | C4⋊C4 |
| LCS | G ▷ C2×C72 ▷ C72 | |
| UCS | C1 ◁ C22 ◁ C2×C4 | |
130 conjugacy classes of subgroups.
| transitive degree | μt(G) = 784 |
Polycyclic presentation:
2^4,7^2|a^2=d,b^2=c,b^a=bd,e^a=e^6,f^a=f^6202 conjugacy classes — view the full character table.
| maximal subgroup of | #13096252, #13096254, #13096261, … |
| maximal normal subgroup of | #13096252, #13096254, #13096261, … |
| maximal quotient of | #13096125, #13096126, #13096132, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(784, 88);