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 | C2×C5×Q8 | C4 |
| prime radical | C2×#713 | C2 |
| cosocle | Dic5 | C24 |
| hypercenter | C2×Q8 | AGL(1,𝔽5) |
| hypocenter | C5 | #568 |
| LCS | G ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C2 ◁ C2×Q8 | |
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)(7,14)(12,20)(13,15)(19,21);(6,9,11,10)(7,13,14,15)(8,16,17,18)(12,19,20,21);(6,8,11,17)(7,12,14,20)(9,18,10,16)(13,21,15,19);(2,3,4,5)(6,7,10,15,11,14,9,13)(8,12,16,19,17,20,18,21)Polycyclic presentation:
2^6,5|a^2=e,b^2=f,c^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 | #10603934, #10604569, #10604571, … |
| maximal normal subgroup of | #10603934, #10604569, #10604571, … |
| maximal quotient of | #10617074, #10617076, #10617078, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 1598);