Finite Group DatabasebetaFinite Group Databasebeta| order | 6976 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
| unambiguous semidirect product | (C2×#89137)⋊Q8 |
| Fitting | C2×Q8×C109 ⋊ C4 |
| Sylow | C109 ⋊ #391 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C22×#89137 |
| commutator, socle | C22×C109 | C22×C4 |
| Fitting | C2×Q8×C109 | C4 |
| hypercenter | C2×Q8 | #89137 |
| hypocenter | C109 | #391 |
| LCS | G ▷ C22×C109 ▷ C109 | |
| UCS | C1 ◁ C22 ◁ C2×Q8 | |
186 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 186 conjugacy classes of subgroups — too many to draw. Only its 64 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Polycyclic presentation:
2^6,109|a^2=f,b^2=def,c^2=e,b^a=bd,c^a=ce,g^a=g^76,g^b=g^33,g^f=g^108298 conjugacy classes — view the full character table.
| maximal subgroup of | #28378128, #28378879, #28378988, … |
| maximal normal subgroup of | #28378128, #28378879, #28378988, … |
| maximal quotient of | #28378209, #28378211, #28378217, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6976, 1502);