Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C5 ⋊ #381 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C22×D5 |
| commutator | C22×C5 | C22×C4 |
| Fitting | C23×C4×C5 | C2 |
| socle, cosocle | C23×C5 | C23 |
| prime radical | C24×C5 | C22 |
| hypercenter | C23×C4 | D5 |
| hypocenter | C5 | #381 |
| LCS | G ▷ C22×C5 ▷ C5 | |
| UCS | C1 ◁ C23 ◁ C23×C4 | |
218 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 3 generators on 21 points:
(1,2,4,5,3)(11,13)(14,16,17,20)(15,19,18,21);(15,18)(19,21);(2,3)(4,5)(6,7,8,9)(10,11)(12,13)(14,15)(16,19)(17,18)(20,21)Polycyclic presentation:
2^6,5|a^2=d,c^2=f,b^a=bf,c^a=ce,g^a=g^492 conjugacy classes — view the full character table.
| maximal subgroup of | #10598107, #10598173, #10598177, … |
| maximal normal subgroup of | #10598107, #10598173, #10598177, … |
| maximal quotient of | #10602324, #10602327, #10602335, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 838);