Finite Group DatabasebetaFinite Group Databasebeta| order | 784 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C22×C72 ⋊ C22 |
| Sylow | C72 ⋊ C2×D4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×D72 |
| commutator, socle, cosocle | C2×C72 | C23 |
| Fitting | C22×C72 | C22 |
| hypercenter | C22 | D72 |
| hypocenter | C72 | C2×D4 |
| LCS | G ▷ C2×C72 ▷ C72 | |
| UCS | C1 ◁ C2 ◁ C22 | |
156 conjugacy classes of subgroups.
| transitive degree | μt(G) = 28 |
Polycyclic presentation:
2^4,7^2|c^a=cd,f^a=f^6,c^b=cd,e^b=e^676 conjugacy classes — view the full character table.
| maximal subgroup of | #13096191, #13096647, #13096655, … |
| maximal normal subgroup of | #13096191, #13096647, #13096655, … |
| maximal quotient of | #13096308, #13096316, #13096318, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(784, 135);