Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C5×#123 ⋊ C2 |
| Sylow | C5 ⋊ #536 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D5 |
| commutator, socle, cosocle | C22×C5 | C24 |
| Fitting | C5×#123 | C2 |
| prime radical | C2×#711 | C2 |
| hypercenter | #123 | D5 |
| hypocenter | C5 | #536 |
| LCS | G ▷ C22×C5 ▷ C5 | |
| UCS | C1 ◁ C22 ◁ #123 | |
270 conjugacy classes of subgroups; only the 103 normal subgroups are shown.
This group has 270 conjugacy classes of subgroups — too many to draw. Only its 103 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 4 generators on 25 points:
(1,2,4,5,3)(7,9)(10,13,14,21)(11,16,17,20)(12,19,18,24)(15,23,22,25);(6,7)(8,9)(12,18)(15,22)(19,24)(23,25);(10,12)(11,15)(13,19)(14,18)(16,23)(17,22)(20,25)(21,24);(2,3)(4,5)(10,11,14,17)(12,15,18,22)(13,20,21,16)(19,25,24,23)Polycyclic presentation:
2^6,5|a^2=f,d^2=f,d^a=df,g^a=g^4,c^b=cf,d^c=de50 conjugacy classes — view the full character table.
| maximal subgroup of | #10599938, #10604532, #10604536, … |
| maximal normal subgroup of | #10599938, #10604532, #10604536, … |
| maximal quotient of | #10605784, #10605787, #10605795, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 1320);