Finite Group DatabasebetaFinite Group Databasebeta| order | 6976 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C109 ⋊ #398 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C22×D109 |
| commutator, socle | C23×C109 | C23 |
| Fitting | C2×C4⋊C4×C109 | C2 |
| hypercenter | C2×C4⋊C4 | D109 |
| hypocenter | C109 | #398 |
| LCS | G ▷ C23×C109 ▷ C109 | |
| UCS | C1 ◁ C23 ◁ C2×C4⋊C4 | |
150 conjugacy classes of subgroups; only the 63 normal subgroups are shown.
This group has 150 conjugacy classes of subgroups — too many to draw. Only its 63 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=e,b^2=d,c^2=d,b^a=bd,c^a=ce,g^a=g^108,c^b=cf1102 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #398×D109, C2×#20294959, #28366274, … |
| maximal normal subgroup of | #398×D109, C2×#20294959, #28366274, … |
| maximal quotient of | #28361284, #28361285, #28361286, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6976, 482);