Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C5 ⋊ #436 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2×C4 | C22×D5 |
| commutator, socle | C22×C5 | C22×C4 |
| Fitting | C4×C5×D4 | C2 |
| prime radical | C2×C5×D4 | C4 |
| cosocle | C2×C4×C5 | C23 |
| hypercenter | C4×D4 | D5 |
| hypocenter | C5 | #436 |
| LCS | G ▷ C22×C5 ▷ C5 | |
| UCS | C1 ◁ C2×C4 ◁ C4×D4 | |
122 conjugacy classes of subgroups.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 3 generators on 17 points:
(1,2,4,5,3)(7,9)(11,14)(15,17);(6,7)(8,9)(11,14)(15,17);(2,3)(4,5)(10,11,12,15,13,14,16,17)Polycyclic presentation:
2^6,5|a^2=d,d^2=f,b^a=bf,c^a=cf,g^a=g^4,c^b=ce68 conjugacy classes — view the full character table.
| maximal subgroup of | #10598731, #10598946, #10598948, … |
| maximal normal subgroup of | #10598731, #10598946, #10598948, … |
| maximal quotient of | #10600241, #10600243, #10600244, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 639);