Finite Group DatabasebetaFinite Group Databasebeta| order | 10688 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C22⋊Q8×C167 ⋊ C2 |
| Sylow | C167 ⋊ #538 |
| 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 | #538 |
| LCS | G ▷ C22×C167 ▷ C167 | |
| UCS | C1 ◁ C22 ◁ C22⋊Q8 | |
250 conjugacy classes of subgroups; only the 95 normal subgroups are shown.
This group has 250 conjugacy classes of subgroups — too many to draw. Only its 95 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=f,b^a=be,c^a=cf,d^a=de,g^a=g^166,c^b=ce,g^b=g^166,g^c=g^1661184 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #36323083, #36323087, #36323117, … |
| maximal normal subgroup of | #36323083, #36323087, #36323117, … |
| maximal quotient of | #36326735, #36326766, #36326795, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(10688, 1139);