Finite Group DatabasebetaFinite Group Databasebeta| order | 784 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C72 ⋊ C2×Q8 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×D72 |
| commutator, socle, cosocle | C2×C72 | C23 |
| Fitting | C4×C72 | C22 |
| prime radical | C2×C7⋊D7 | C22 |
| hypercenter | C4 | D72 |
| hypocenter | C72 | C2×Q8 |
| LCS | G ▷ C2×C72 ▷ C72 | |
| UCS | C1 ◁ C2 ◁ C4 | |
100 conjugacy classes of subgroups.
| transitive degree | μt(G) = 56 |
Polycyclic presentation:
2^4,7^2|a^2=d,b^2=d,c^2=d,c^a=cd,f^a=f^6,c^b=cd,e^b=e^676 conjugacy classes — view the full character table.
| maximal subgroup of | #13096181, #13096183, #13096234, … |
| maximal normal subgroup of | #13096181, #13096183, #13096234, … |
| maximal quotient of | #13096251, #13096254, #13096272, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(784, 121);