Finite Group DatabasebetaFinite Group Databasebeta| order | 10688 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C167 ⋊ #389 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C22×D167 |
| commutator | C22×C167 | C22×C4 |
| Fitting | #95×C167 | C2 |
| socle, prime radical, cosocle | C23×C167 | C23 |
| hypercenter | #95 | D167 |
| hypocenter | C167 | #389 |
| LCS | G ▷ C22×C167 ▷ C167 | |
| UCS | C1 ◁ C23 ◁ #95 | |
170 conjugacy classes of subgroups; only the 77 normal subgroups are shown.
This group has 170 conjugacy classes of subgroups — too many to draw. Only its 77 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Polycyclic presentation:
2^6,167|a^2=f,b^2=f,c^2=e,b^a=bd,c^a=ce,g^b=g^166,g^c=g^1661688 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #389×D167, C2×#24923166, #36326598, … |
| maximal normal subgroup of | #389×D167, C2×#24923166, #36326598, … |
| maximal quotient of | Dic167⋊#95, #36322369, #36322372, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(10688, 446);