Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C5×C4⋊C4 ⋊ C4 |
| Sylow | C5 ⋊ #389 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C22×AGL(1,𝔽5) |
| commutator, socle | C22×C5 | C22×C4 |
| Fitting | C5×C4⋊C4 | C4 |
| prime radical, cosocle | C22×D5 | C23 |
| hypercenter | C4⋊C4 | AGL(1,𝔽5) |
| hypocenter | C5 | #389 |
| LCS | G ▷ C22×C5 ▷ C5 | |
| UCS | C1 ◁ C22 ◁ C4⋊C4 | |
170 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 170 conjugacy classes of subgroups — too many to draw. Only its 64 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 13 |
| transitive degree | μt(G) = 80 |
A minimal faithful permutation representation — 3 generators on 13 points:
(1,2,3,5,4)(7,9)(10,11,12,13);(6,7,8,9);(2,3,4,5)(11,13)Polycyclic presentation:
2^6,5|a^2=d,b^2=f,c^2=e,c^a=ce,g^a=g^2,c^b=cf,g^d=g^438 conjugacy classes — view the full character table.
| maximal subgroup of | #10603957, #10603958, #10603963, … |
| maximal normal subgroup of | #10603957, #10603958, #10603963, … |
| maximal quotient of | #10603796, #10603797, #10603798, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 1050);