Finite Group DatabasebetaFinite Group Databasebeta| order | 8768 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | #131×C137 ⋊ C2 |
| Sylow | C137 ⋊ #443 |
| subgroup | G/H | |
|---|---|---|
| center | C8 | C22⋊D137 |
| commutator | C4×C137 | C22×C4 |
| Frattini | C2×C4 | C22×D137 |
| Fitting | #131×C137 | C2 |
| socle | C2×C137 | C4×D4 |
| hypercenter | #131 | D137 |
| hypocenter | C137 | #443 |
| LCS | G ▷ C4×C137 ▷ C2×C137 ▷ C137 | |
| UCS | C1 ◁ C8 ◁ C2×C8 ◁ #131 | |
106 conjugacy classes of subgroups; only the 47 normal subgroups are shown.
This group has 106 conjugacy classes of subgroups — too many to draw. Only its 47 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,e^2=f,b^a=bd,d^a=df,d^b=df,g^b=g^1361388 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #31782623, #31782624, #31782628, … |
| maximal normal subgroup of | #31782623, #31782624, #31782628, … |
| maximal quotient of | C8×#14555495, C8×#14555498, C8×#14555497, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8768, 664);