Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| isoclinism | isoclinic class of #700 |
| automorphism | Aut(G) = C4×D4×AGL(1,𝔽5) (order 640) · Inn(G) = C2×AGL(1,𝔽5) · Out(G) = C22×C4 |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Fitting | C5×C16 ⋊ C4 |
| Sylow | C5 ⋊ #346 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C8 | C2×AGL(1,𝔽5) |
| commutator, socle | C2×C5 | C4×C8 |
| Fitting | C5×C16 | C4 |
| prime radical | C2×D5 | C2×C8 |
| cosocle | C8×D5 | C22 |
| hypercenter | C16 | AGL(1,𝔽5) |
| hypocenter | C5 | #346 |
| LCS | G ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C8 ◁ C16 | |
54 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 80 |
A minimal faithful permutation representation — 2 generators on 21 points:
(1,2,3,5,4)(6,7,8,12,9,13,14,19,10,11,15,17,16,18,20,21);(2,3,4,5)(7,11)(12,17)(13,18)(19,21)Polycyclic presentation:
2^6,5|a^2=c,b^2=d,d^2=e,e^2=f,b^a=bf,g^a=g^2,g^c=g^456 conjugacy classes — view the full character table.
| maximal subgroup of | #10598487, #10598488, C2×#64844, … |
| maximal normal subgroup of | #10598487, #10598488, C2×#64844, … |
| maximal quotient of | #10598487, #10598488, #10598636, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 182);