Finite Group DatabasebetaFinite Group Databasebeta| order | 10688 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C167 ⋊ #554 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D167 |
| commutator, socle, cosocle | C22×C167 | C24 |
| Fitting | C4×D4×C167 | C2 |
| prime radical | C2×D4×C167 | C22 |
| hypercenter | C4×D4 | D167 |
| hypocenter | C167 | #554 |
| LCS | G ▷ C22×C167 ▷ C167 | |
| UCS | C1 ◁ C22 ◁ C4×D4 | |
228 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 228 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|a^2=e,b^2=ef,d^2=e,b^a=be,c^a=cf,d^b=de,g^b=g^1661685 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #36323208, #36323212, #36323270, … |
| maximal normal subgroup of | #36323208, #36323212, #36323270, … |
| maximal quotient of | #36327013, D4×C4⋊Dic167, #36327043, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(10688, 1256);