Finite Group DatabasebetaFinite Group Databasebeta| order | 10688 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C167 ⋊ #395 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C22×D167 |
| commutator, socle, prime radical, cosocle | C23×C167 | C23 |
| Fitting | C2×C4⋊C4×C167 | C2 |
| hypercenter | C2×C4⋊C4 | D167 |
| hypocenter | C167 | #395 |
| LCS | G ▷ C23×C167 ▷ C167 | |
| UCS | C1 ◁ C23 ◁ C2×C4⋊C4 | |
182 conjugacy classes of subgroups.
Polycyclic presentation:
2^6,167|a^2=d,b^2=d,c^2=d,b^a=bd,c^a=ce,c^b=cf,g^b=g^1661682 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #36322167, #36322169, #36323101, … |
| maximal normal subgroup of | #36322167, #36322169, #36323101, … |
| maximal quotient of | #36322389, #36322391, #36322401, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(10688, 469);