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 ⋊ #435 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2×C4 | C22×D5 |
| commutator, socle | C22×C5 | C22×C4 |
| Fitting | C5×C4⋊C8 | C2 |
| prime radical | C2×D20 | C4 |
| cosocle | C2×C4×C5 | C23 |
| hypercenter | C4⋊C8 | D5 |
| hypocenter | C5 | #435 |
| LCS | G ▷ C22×C5 ▷ C5 | |
| UCS | C1 ◁ C2×C4 ◁ C4⋊C8 | |
124 conjugacy classes of subgroups; only the 51 normal subgroups are shown.
This group has 124 conjugacy classes of subgroups — too many to draw. Only its 51 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| 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)(10,12,13,16,14,17,11,15);(6,7,8,9)(10,11,14,13)(12,16,17,15);(2,3)(4,5)(7,9)Polycyclic presentation:
2^6,5|b^2=e,c^2=d,d^2=f,b^a=bef,g^a=g^4,c^b=ce68 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#65125, #10610735, #10610746, … |
| maximal normal subgroup of | C2×#65125, #10610735, #10610746, … |
| maximal quotient of | #10599312, #10599313, #10599314, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 463);