Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| isoclinism | isoclinic class of AGL(1,𝔽5) |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C16 · Schur multiplier (H2) = C4 |
No aliases for this group.
| derived, Sylow | C5 ⋊ C4×C16 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C8 | AGL(1,𝔽5) |
| commutator, hypocenter | C5 | C4×C16 |
| Frattini | C2×C4 | C2×AGL(1,𝔽5) |
| Fitting | C2×C5×C8 | C4 |
| socle, prime radical | C22×C5 | C2×C8 |
| cosocle | C2×#168 | C22 |
| LCS | G ▷ C5 | |
| UCS | C1 ◁ C2×C8 | |
58 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 2 generators on 25 points:
(1,2,3,5,4)(6,7,8,9)(10,12,20,22,14,21,15,13)(11,16,23,25,18,24,19,17);(2,3,4,5)(10,11,13,17,15,19,21,24,14,18,22,25,20,23,12,16)Polycyclic presentation:
2^6,5|a^2=c,b^2=de,c^2=e,e^2=f,g^a=g^2,g^c=g^480 conjugacy classes — view the full character table.
| maximal subgroup of | #10598609, #10598610, C16×#170, … |
| maximal normal subgroup of | #10598609, #10598610, C16×#170, … |
| maximal quotient of | C8×#674, #10598514, #10598652, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 223);