Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Sylow | C5 ⋊ #519 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D5 |
| commutator | C2×C5 | C23×C4 |
| Fitting | C5×#117 | C2 |
| socle, cosocle | C22×C5 | C24 |
| prime radical | C23×C5 | C23 |
| hypercenter | #117 | D5 |
| hypocenter | C5 | #519 |
| LCS | G ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C22 ◁ #117 | |
266 conjugacy classes of subgroups.
| 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)(6,7,8,9)(12,19)(15,23)(18,25)(22,24);(10,13,14,21)(11,16,17,20)(12,18,19,25)(15,22,23,24);(10,12)(11,15)(13,18)(14,19)(16,22)(17,23)(20,24)(21,25);(2,3)(4,5)(10,11,14,17)(12,15,19,23)(13,20,21,16)(18,24,25,22)Polycyclic presentation:
2^6,5|a^2=f,c^2=f,d^2=e,c^a=cf,g^a=g^4,d^b=df74 conjugacy classes — view the full character table.
| maximal subgroup of | #10599500, #10599501, #10599557, … |
| maximal normal subgroup of | #10599500, #10599501, #10599557, … |
| maximal quotient of | #10605114, #10605117, #10605119, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 1188);