Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C5×C22⋊C8 ⋊ C2 |
| Sylow | C5 ⋊ #407 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2×C4 | C22×D5 |
| commutator, socle | C22×C5 | C22×C4 |
| Fitting | C5×C22⋊C8 | C2 |
| prime radical | C23×D5 | C4 |
| cosocle | C2×C4×C5 | C23 |
| hypercenter | C22⋊C8 | D5 |
| hypocenter | C5 | #407 |
| LCS | G ▷ C22×C5 ▷ C5 | |
| UCS | C1 ◁ C2×C4 ◁ C22⋊C8 | |
190 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 190 conjugacy classes of subgroups — too many to draw. Only its 61 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) = 80 |
A minimal faithful permutation representation — 3 generators on 17 points:
(1,2,4,5,3)(7,9)(10,11,12,15,13,14,16,17);(6,7)(8,9);(2,3)(4,5)(11,14)(15,17)Polycyclic presentation:
2^6,5|c^2=d,d^2=f,c^a=cf,g^a=g^4,c^b=ce68 conjugacy classes — view the full character table.
| maximal subgroup of | #10598553, #10598554, #10598555, … |
| maximal normal subgroup of | #10598553, #10598554, #10598555, … |
| maximal quotient of | #10598895, #10598896, #10598913, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 353);