Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C5 ⋊ #435 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C22×AGL(1,𝔽5) |
| commutator, socle | C22×C5 | C22×C4 |
| Fitting, prime radical | C2×C5×D4 | C4 |
| cosocle | C2×Dic5 | C23 |
| hypercenter | C2×D4 | AGL(1,𝔽5) |
| hypocenter | C5 | #435 |
| LCS | G ▷ C22×C5 ▷ C5 | |
| UCS | C1 ◁ C22 ◁ C2×D4 | |
124 conjugacy classes of subgroups; only the 48 normal subgroups are shown.
This group has 124 conjugacy classes of subgroups — too many to draw. Only its 48 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 — 3 generators on 17 points:
(1,2,3,5,4)(7,9)(11,14)(15,17);(6,7)(8,9);(2,3,4,5)(10,11,12,15,13,14,16,17)Polycyclic presentation:
2^6,5|a^2=d,d^2=f,c^a=cf,g^a=g^2,c^b=ce,g^d=g^438 conjugacy classes — view the full character table.
| maximal subgroup of | D5⋊#435, #10617150, #10617154, … |
| maximal normal subgroup of | D5⋊#435, #10617150, #10617154, … |
| maximal quotient of | #10604226, #10604227, #10604230, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 1111);