Finite Group DatabasebetaFinite Group Databasebeta| order | 10688 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C2×C22⋊C4×C167 ⋊ C2 |
| Sylow | C167 ⋊ #392 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C22×D167 |
| commutator, socle, cosocle | C23×C167 | C23 |
| Fitting | C2×C22⋊C4×C167 | C2 |
| hypercenter | C2×C22⋊C4 | D167 |
| hypocenter | C167 | #392 |
| LCS | G ▷ C23×C167 ▷ C167 | |
| UCS | C1 ◁ C23 ◁ C2×C22⋊C4 | |
322 conjugacy classes of subgroups; only the 67 normal subgroups are shown.
This group has 322 conjugacy classes of subgroups — too many to draw. Only its 67 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Polycyclic presentation:
2^6,167|b^a=bd,c^a=ce,c^b=cf,g^b=g^166,g^c=g^1661682 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #36322134, #36322136, #36323056, … |
| maximal normal subgroup of | #36322134, #36322136, #36323056, … |
| maximal quotient of | #36322377, #36322378, #36322382, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(10688, 458);