Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C2×#4041 |
| Fitting | C2×C5×C4⋊C4 ⋊ C2 |
| Sylow | C5 ⋊ C2×#121 |
| subgroup | G/H | |
|---|---|---|
| center | C23 | C22×D5 |
| commutator, cosocle | C22×C5 | C24 |
| Frattini | C22 | C23×D5 |
| Fitting | C2×C5×C4⋊C4 | C2 |
| socle | C23×C5 | C23 |
| hypercenter | C2×C4⋊C4 | D5 |
| hypocenter | C5 | C2×#121 |
| LCS | G ▷ C22×C5 ▷ C5 | |
| UCS | C1 ◁ C23 ◁ C2×C4⋊C4 | |
426 conjugacy classes of subgroups; only the 135 normal subgroups are shown.
This group has 426 conjugacy classes of subgroups — too many to draw. Only its 135 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) = 160 |
A minimal faithful permutation representation — 4 generators on 15 points:
(3,4,5,6)(8,10)(11,12,14,15,13);(7,8,9,10);(1,2)(4,6)(12,13)(14,15);(1,2)68 conjugacy classes — view the full character table.
| maximal subgroup of | #10599615, #10599616, #10599673, … |
| maximal normal subgroup of | #10599615, #10599616, #10599673, … |
| maximal quotient of | C2×#64959, C2×#64961, C2×#64963, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 1178);