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 ⋊ #548 |
| 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 | #548 |
| LCS | G ▷ C22×C167 ▷ C167 | |
| UCS | C1 ◁ C22 ◁ C22×Q8 | |
290 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 290 conjugacy classes of subgroups — too many to draw. Only its 115 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^2=e,d^2=e,b^a=be,c^a=cf,g^a=g^166,d^b=de1685 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #36335626, #36335633, #36335687, … |
| maximal normal subgroup of | #36335626, #36335633, #36335687, … |
| maximal quotient of | Q8×C22⋊Dic167, #36326976, #36327056, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(10688, 1210);