Finite Group DatabasebetaFinite Group Databasebeta| order | 10688 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C22⋊Q8×C167 ⋊ C2 |
| Sylow | C167 ⋊ #547 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D167 |
| commutator, socle, cosocle | C22×C167 | C24 |
| Fitting | C22⋊Q8×C167 | C2 |
| hypercenter | C22⋊Q8 | D167 |
| hypocenter | C167 | #547 |
| LCS | G ▷ C22×C167 ▷ C167 | |
| UCS | C1 ◁ C22 ◁ C22⋊Q8 | |
292 conjugacy classes of subgroups; only the 105 normal subgroups are shown.
This group has 292 conjugacy classes of subgroups — too many to draw. Only its 105 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=e,c^2=e,b^a=be,c^a=cf,d^b=de,g^b=g^166,g^d=g^1661187 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #36335660, #36335661, #36335667, … |
| maximal normal subgroup of | #36335660, #36335661, #36335667, … |
| maximal quotient of | #36326966, #36326967, #36327004, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(10688, 1201);