Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Sylow | C5 ⋊ #568 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C23×AGL(1,𝔽5) |
| commutator, socle | C2×C5 | C23×C4 |
| Fitting | C5×#41 | C4 |
| prime radical | D5×#41 | C2 |
| cosocle | Dic5 | C24 |
| hypercenter | #41 | AGL(1,𝔽5) |
| hypocenter | C5 | #568 |
| LCS | G ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C2 ◁ #41 | |
258 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 80 |
A minimal faithful permutation representation — 4 generators on 21 points:
(1,2,3,5,4)(6,9,11,10)(7,13,14,18)(8,16,15,17)(12,20,19,21);(8,15)(12,19)(16,17)(20,21);(6,8)(7,12)(9,16)(10,17)(11,15)(13,20)(14,19)(18,21);(2,3,4,5)(6,7,10,13,11,14,9,18)(8,12,17,20,15,19,16,21)Polycyclic presentation:
2^6,5|a^2=e,d^2=f,e^2=f,d^a=df,g^a=g^2,c^b=cf,g^e=g^444 conjugacy classes — view the full character table.
| maximal subgroup of | #10604007, #10604008, #10604865, … |
| maximal normal subgroup of | #10604007, #10604008, #10604865, … |
| maximal quotient of | #10617106, #10617108, #10617110, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 1602);