Finite Group DatabasebetaFinite Group Databasebeta| order | 6976 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C4×Q8×C109 ⋊ C2 |
| Sylow | C109 ⋊ #548 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D109 |
| commutator, socle | C22×C109 | C24 |
| Fitting | C4×Q8×C109 | C2 |
| hypercenter | C4×Q8 | D109 |
| hypocenter | C109 | #548 |
| LCS | G ▷ C22×C109 ▷ C109 | |
| UCS | C1 ◁ C22 ◁ C4×Q8 | |
290 conjugacy classes of subgroups; only the 113 normal subgroups are shown.
This group has 290 conjugacy classes of subgroups — too many to draw. Only its 113 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Polycyclic presentation:
2^6,109|b^2=e,d^2=e,b^a=be,c^a=cf,g^a=g^108,d^b=de,g^c=g^1081105 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #28374522, #28374529, #28374582, … |
| maximal normal subgroup of | #28374522, #28374529, #28374582, … |
| maximal quotient of | Q8×#13182006, #28365870, #28365953, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6976, 1211);