Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C5×C4⋊C8 ⋊ C2 |
| Sylow | C5 ⋊ #424 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2×C4 | C22×D5 |
| commutator, socle | C22×C5 | C22×C4 |
| Fitting | C5×C4⋊C8 | C2 |
| prime radical | C22×D5 | C2×C4 |
| cosocle | C2×C4×C5 | C23 |
| hypercenter | C4⋊C8 | D5 |
| hypocenter | C5 | #424 |
| LCS | G ▷ C22×C5 ▷ C5 | |
| UCS | C1 ◁ C2×C4 ◁ C4⋊C8 | |
114 conjugacy classes of subgroups; only the 55 normal subgroups are shown.
This group has 114 conjugacy classes of subgroups — too many to draw. Only its 55 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| 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)(10,12,13,18,14,19,20,24)(11,16,17,23,15,21,22,25);(6,7,8,9)(11,15)(16,21)(17,22)(23,25);(2,3)(4,5)(10,11)(12,16)(13,17)(14,15)(18,23)(19,21)(20,22)(24,25)Polycyclic presentation:
2^6,5|b^2=e,c^2=d,d^2=f,b^a=bf,g^a=g^4,c^b=ce68 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#65128, #10610743, #10610747, … |
| maximal normal subgroup of | C2×#65128, #10610743, #10610747, … |
| maximal quotient of | #10599325, #10599326, #10599327, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 466);