Finite Group DatabasebetaFinite Group Databasebeta| order | 8768 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | #97×C137 ⋊ C2 |
| Sylow | C137 ⋊ #440 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×D548 |
| commutator | C4×C137 | C22×C4 |
| Frattini | C2×C4 | C22×D137 |
| Fitting | #97×C137 | C2 |
| socle | C22×C137 | C2×D4 |
| hypercenter | #97 | D137 |
| hypocenter | C137 | #440 |
| LCS | G ▷ C4×C137 ▷ C2×C137 ▷ C137 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ #97 | |
120 conjugacy classes of subgroups; only the 51 normal subgroups are shown.
This group has 120 conjugacy classes of subgroups — too many to draw. Only its 51 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Polycyclic presentation:
2^6,137|a^2=e,c^2=e,d^2=f,b^a=bd,c^a=cf,d^a=df,g^a=g^136,d^b=df,g^b=g^1361382 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #440×D137, C2×#22577505, #31789584, … |
| maximal normal subgroup of | #440×D137, C2×#22577505, #31789584, … |
| maximal quotient of | #31779603, #31779648, #31779669, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8768, 651);