Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C5 ⋊ #408 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2×C4 | C22×D5 |
| commutator, socle | C22×C5 | C22×C4 |
| Fitting, prime radical | C2×C5×#41 | C2 |
| cosocle | C2×C4×C5 | C23 |
| hypercenter | C2×#41 | D5 |
| hypocenter | C5 | #408 |
| LCS | G ▷ C22×C5 ▷ C5 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#41 | |
158 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 3 generators on 25 points:
(1,2,4,5,3)(7,9)(12,18)(15,21)(19,24)(22,25);(10,12)(11,15)(13,19)(14,18)(16,22)(17,21)(20,24)(23,25);(2,3)(4,5)(6,7)(8,9)(10,11,13,16,14,17,20,23)(12,15,19,22,18,21,24,25)Polycyclic presentation:
2^6,5|a^2=d,d^2=f,c^a=ce,g^a=g^4,c^b=cf68 conjugacy classes — view the full character table.
| maximal subgroup of | #10598187, #10598188, #10598191, … |
| maximal normal subgroup of | #10598187, #10598188, #10598191, … |
| maximal quotient of | #10602372, #10602375, #10602376, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 861);