Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#4144 |
| Fitting | C22×C5×D4 ⋊ C2 |
| Sylow | C5 ⋊ C2×H(5,𝔽2) |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C23×D5 |
| commutator, cosocle | C2×C5 | C25 |
| Frattini | C2 | C24×D5 |
| Fitting | C22×C5×D4 | C2 |
| socle | C22×C5 | C24 |
| hypercenter | C22×D4 | D5 |
| hypocenter | C5 | C2×H(5,𝔽2) |
| LCS | G ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C22 ◁ C22×D4 | |
898 conjugacy classes of subgroups; only the 447 normal subgroups are shown.
This group has 898 conjugacy classes of subgroups — too many to draw. Only its 447 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 80 |
A minimal faithful permutation representation — 5 generators on 15 points:
(1,2)(3,4,6,7,5)(8,10)(9,13)(11,14)(12,15);(8,9)(10,13)(11,12)(14,15);(9,12)(13,15);(4,5)(6,7)(10,14)(13,15);(1,2)74 conjugacy classes — view the full character table.
| maximal subgroup of | #10600391, #10600455, #10600826, … |
| maximal normal subgroup of | #10600391, #10600455, #10600826, … |
| maximal quotient of | C2×#65817, C2×#65829, C2×#65820, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 1614);