Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C53×#322 ⋊ C2 |
| Sylow | C53 ⋊ #1580 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×D212 |
| commutator | C4×C53 | C22×C8 |
| Frattini | C42 | C22×D53 |
| Fitting | C53×#322 | C2 |
| socle | C22×C53 | C4×D4 |
| hypercenter | #322 | D53 |
| hypocenter | C53 | #1580 |
| LCS | G ▷ C4×C53 ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C2×C4 ◁ C4×C8 ◁ #322 | |
192 conjugacy classes of subgroups; only the 73 normal subgroups are shown.
This group has 192 conjugacy classes of subgroups — too many to draw. Only its 73 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=e,c^2=ef,d^2=f,e^2=g,b^a=bd,c^a=cf,d^a=df,h^a=h^52,d^b=df,h^b=h^521084 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #27295963, #27295967, #27295975, … |
| maximal normal subgroup of | #27295963, #27295967, #27295975, … |
| maximal quotient of | #27274870, #27275336, #27278020, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 3335);