Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
| direct product | Q8×C22⋊D5 |
| Fitting | C22×C5×Q8 ⋊ C2 |
| Sylow | C5 ⋊ D4×Q8 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D5 |
| commutator, socle, cosocle | C22×C5 | C24 |
| Fitting | C22×C5×Q8 | C2 |
| prime radical | C2×C22⋊D5 | C22 |
| hypercenter | C22×Q8 | D5 |
| hypocenter | C5 | D4×Q8 |
| LCS | G ▷ C22×C5 ▷ C5 | |
| UCS | C1 ◁ C22 ◁ C22×Q8 | |
280 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 280 conjugacy classes of subgroups — too many to draw. Only its 123 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 4 generators on 17 points:
(1,2,4,5,3)(7,9);(2,3)(4,5)(6,7)(8,9);(10,12,13,17)(11,14,15,16);(10,11,13,15)(12,16,17,14)65 conjugacy classes — view the full character table.
| maximal subgroup of | #10610536, #10610537, #10610541, … |
| maximal normal subgroup of | #10610536, #10610537, #10610541, … |
| maximal quotient of | #10613893, #10613902, #10613904, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 1487);