Finite Group DatabasebetaFinite Group Databasebeta| order | 6976 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C109 ⋊ #389 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C22×D109 |
| commutator | C22×C109 | C22×C4 |
| Fitting | #95×C109 | C2 |
| socle | C23×C109 | C23 |
| hypercenter | #95 | D109 |
| hypocenter | C109 | #389 |
| LCS | G ▷ C22×C109 ▷ C109 | |
| UCS | C1 ◁ C23 ◁ #95 | |
170 conjugacy classes of subgroups; only the 77 normal subgroups are shown.
This group has 170 conjugacy classes of subgroups — too many to draw. Only its 77 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=f,c^2=e,b^a=bd,c^a=ce,g^b=g^108,g^c=g^1081108 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #389×D109, C2×#20294923, #28365492, … |
| maximal normal subgroup of | #389×D109, C2×#20294923, #28365492, … |
| maximal quotient of | #28361260, #28361263, #28361266, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6976, 446);