Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| isoclinism | isoclinic class of #64686 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Sylow | C5 ⋊ #347 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | C4×Dic5 |
| commutator | C4×C5 | C42 |
| Frattini | C2×C8 | C2×D5 |
| Fitting | C5×M32 | C2 |
| socle | C2×C5 | #97 |
| prime radical | C22×C5 | M16 |
| cosocle | C2×C5×C8 | C22 |
| hypercenter | M32 | D5 |
| hypocenter | C5 | #347 |
| LCS | G ▷ C4×C5 ▷ C5 | |
| UCS | C1 ◁ C4 ◁ M32 | |
46 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 80 |
A minimal faithful permutation representation — 2 generators on 21 points:
(1,2,4,5,3)(6,7,8,13,9,11,15,18,10,12,14,19,16,17,21,20);(2,3)(4,5)(7,11,12,17)(8,14)(13,20,19,18)(15,21)Polycyclic presentation:
2^6,5|a^2=c,b^2=d,d^2=e,e^2=f,b^a=be,d^a=df,g^a=g^4,c^b=cf62 conjugacy classes — view the full character table.
| maximal subgroup of | D5×#347, #10599880, #10599907, … |
| maximal normal subgroup of | D5×#347, #10599880, #10599907, … |
| maximal quotient of | #10598002, #10598003, #10598302, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 70);