Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C53 ⋊ #2984 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×#6157 |
| commutator, socle | C22×C53 | C23×C4 |
| Fitting | C4×Q8×C53 | C4 |
| hypercenter | C4×Q8 | #6157 |
| hypocenter | C53 | #2984 |
| LCS | G ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C22 ◁ C4×Q8 | |
364 conjugacy classes of subgroups; only the 158 normal subgroups are shown.
This group has 364 conjugacy classes of subgroups — too many to draw. Only its 158 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=f,b^2=e,c^2=e,f^2=g,b^a=be,d^a=de,h^a=h^30,c^b=ceg,h^d=h^52,h^f=h^52310 conjugacy classes — view the full character table.
| maximal subgroup of | #28304537, #28304539, #28304545, … |
| maximal normal subgroup of | #28304537, #28304539, #28304545, … |
| maximal quotient of | #28318706, #28318708, #28318887, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 21411);