Finite Group DatabasebetaFinite Group Databasebeta| order | 10688 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| Fitting | C2×#41×C167 ⋊ C2 |
| Sylow | C167 ⋊ #585 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | C23×D167 |
| commutator, socle, cosocle | C2×C167 | C25 |
| Frattini | C2 | C24×D167 |
| Fitting | C2×#41×C167 | C2 |
| hypercenter | C2×#41 | D167 |
| hypocenter | C167 | #585 |
| LCS | G ▷ C2×C167 ▷ C167 | |
| UCS | C1 ◁ C4 ◁ C2×#41 | |
810 conjugacy classes of subgroups; only the 443 normal subgroups are shown.
This group has 810 conjugacy classes of subgroups — too many to draw. Only its 443 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Polycyclic presentation:
2^6,167|e^2=f,b^a=bf,d^a=df,c^b=cf,g^d=g^1661694 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #36324087, #36324088, #36324100, … |
| maximal normal subgroup of | #36324087, #36324088, #36324100, … |
| maximal quotient of | #36337337, #36337339, #36337341, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(10688, 1385);