Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| isoclinism | isoclinic class of Dic10 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C5 ⋊ #346 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2×C8 | C2×D5 |
| commutator | C2×C5 | C4×C8 |
| Fitting | C4×C5×C8 | C2 |
| socle, prime radical | C22×C5 | C2×C8 |
| cosocle | C2×C5×C8 | C22 |
| hypercenter | C4×C8 | D5 |
| hypocenter | C5 | #346 |
| LCS | G ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C2×C8 ◁ C4×C8 | |
54 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 2 generators on 25 points:
(1,2,4,5,3)(6,7,8,9)(11,15)(16,21)(17,22)(23,25);(2,3)(4,5)(10,11,12,16,13,17,18,23,14,15,19,21,20,22,24,25)Polycyclic presentation:
2^6,5|a^2=c,b^2=d,c^2=e,e^2=f,b^a=bf,g^a=g^4104 conjugacy classes — view the full character table.
| maximal subgroup of | #10597933, #10597955, #10597956, … |
| maximal normal subgroup of | #10597933, #10597955, #10597956, … |
| maximal quotient of | #10597928, #10597929, #10598010, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 19);