Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| Fitting | #124×C53 ⋊ C4 |
| Sylow | C53 ⋊ #2372 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×#6157 |
| commutator, socle | C22×C53 | C23×C4 |
| Fitting | #124×C53 | C4 |
| hypercenter | #124 | #6157 |
| hypocenter | C53 | #2372 |
| LCS | G ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C22 ◁ #124 | |
560 conjugacy classes of subgroups; only the 160 normal subgroups are shown.
This group has 560 conjugacy classes of subgroups — too many to draw. Only its 160 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Polycyclic presentation:
2^7,53|a^2=g,b^2=eg,c^2=e,d^2=f,b^a=be,c^a=cf,h^a=h^23,d^b=de,h^b=h^30,h^g=h^52232 conjugacy classes — view the full character table.
| maximal subgroup of | #28305174, #28305176, #28305177, … |
| maximal normal subgroup of | #28305174, #28305176, #28305177, … |
| maximal quotient of | #28312803, #28312821, #28312837, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 21046);