Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1402 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C53 · Schur multiplier (H2) = C2 |
| direct product | C53×#1403 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C53 | AΣL(1,𝔽16) |
| commutator | C2×D4 | C2×C4×C53 |
| Frattini, cosocle | #124 | C22×C53 |
| prime radical | H(5,𝔽2)×C53 | C4 |
| LCS | G ▷ C2×D4 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C53 ◁ C22×C53 ◁ C2×C4×C53 ◁ #124×C53 ◁ G | |
112 conjugacy classes of subgroups; only the 28 normal subgroups are shown.
This group has 112 conjugacy classes of subgroups — too many to draw. Only its 28 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
901 conjugacy classes — view the full character table.
| maximal subgroup of | C2×C53×#1403, C53×#13564, C53×#13565, … |
| maximal normal subgroup of | C2×C53×#1403, C53×#13564, C53×#13565, … |
| maximal quotient of | C53×#7344, C53×#7375, C53×#7409, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 135);