Finite Group DatabasebetaFinite Group Databasebeta| order | 8768 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| Sylow | C137 ⋊ #388 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C22×D137 |
| commutator | C22×C137 | C22×C4 |
| Fitting | C2×C22⋊C4×C137 | C2 |
| socle | C23×C137 | C23 |
| hypercenter | C2×C22⋊C4 | D137 |
| hypocenter | C137 | #388 |
| LCS | G ▷ C22×C137 ▷ C137 | |
| UCS | C1 ◁ C23 ◁ C2×C22⋊C4 | |
190 conjugacy classes of subgroups; only the 75 normal subgroups are shown.
This group has 190 conjugacy classes of subgroups — too many to draw. Only its 75 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=f,b^2=df,c^2=d,b^a=bd,c^a=ce,g^a=g^136,g^b=g^1361388 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #388×D137, C2×#22577296, #31783231, … |
| maximal normal subgroup of | #388×D137, C2×#22577296, #31783231, … |
| maximal quotient of | #31779014, #31779023, #31779026, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8768, 442);