Finite Group DatabasebetaFinite Group Databasebeta| order | 17745 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C5 · Schur multiplier (H2) = C1 |
| direct product | C5×#16813064 |
| Fitting, Hall | C5×C7×C169 ⋊ C3 |
| derived, Hall | C7×C169 ⋊ C3×C5 |
| Hall | C5×C169 ⋊ ΓL(1,𝔽8) |
| Sylow | C169 ⋊ C5×ΓL(1,𝔽8) |
| Hall | C5×C7 ⋊ C169⋊C3 |
| Sylow | C7 ⋊ C5×C169⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C5 | #16813064 |
| commutator, cosocle, hypocenter | C7×C169 | C3×C5 |
| Frattini | C13 | C5×#63271 |
| Fitting | C5×C7×C169 | C3 |
| socle | C5×C7×C13 | C13⋊C3 |
| LCS | G ▷ C7×C169 | |
| UCS | C1 ◁ C5 | |
24 conjugacy classes of subgroups.
| transitive degree | μt(G) = 5915 |
1985 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C2×C5×#16813064, C5×#20296859, C5×#20296863, … |
| maximal normal subgroup of | C2×C5×#16813064, C5×#20296859, C5×#20296863, … |
| maximal quotient of | C2×C5×#16813064 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(17745, 4);