Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| isoclinism | isoclinic class of #686 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
| direct product | C2×#4096 |
| Fitting | C2×C5×#41 ⋊ C2 |
| Sylow | C5 ⋊ C2×#135 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22⋊D5 |
| commutator, cosocle | C4×C5 | C24 |
| Frattini | C4 | C23×D5 |
| Fitting | C2×C5×#41 | C2 |
| socle | C22×C5 | C2×D4 |
| hypercenter | C2×#41 | D5 |
| hypocenter | C5 | C2×#135 |
| LCS | G ▷ C4×C5 ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C2×#41 | |
266 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 266 conjugacy classes of subgroups — too many to draw. Only its 111 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 4 generators on 23 points:
(3,4,6,7,5)(9,14)(11,19)(13,20)(15,16)(17,23)(21,22);(1,2)(8,10,12,18)(9,13,16,22)(11,17,19,23)(14,20,15,21);(4,5)(6,7)(8,9)(10,13)(11,14)(12,16)(15,19)(17,20)(18,22)(21,23);(1,2)68 conjugacy classes — view the full character table.
| maximal subgroup of | #10600792, #10600793, #10600804, … |
| maximal normal subgroup of | #10600792, #10600793, #10600804, … |
| maximal quotient of | C2×#65287, C2×#65291, C2×#65292, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 1493);