Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C5×#100 ⋊ C2 |
| Sylow | C5 ⋊ #472 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #4030 |
| commutator, cosocle | C2×C4×C5 | C23 |
| Frattini | C2×C4 | C22×D5 |
| Fitting | C5×#100 | C2 |
| socle | C2×C5 | #121 |
| hypercenter | #100 | D5 |
| hypocenter | C5 | #472 |
| LCS | G ▷ C2×C4×C5 ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #100 | |
108 conjugacy classes of subgroups; only the 35 normal subgroups are shown.
This group has 108 conjugacy classes of subgroups — too many to draw. Only its 35 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 80 |
A minimal faithful permutation representation — 3 generators on 21 points:
(1,2,4,5,3)(7,12)(8,15)(9,17)(11,18)(13,14)(19,20);(6,8,9,16,10,15,17,21)(7,11,13,19,14,20,12,18);(2,3)(4,5)(6,7)(8,11)(9,13)(10,14)(12,17)(15,20)(16,19)(18,21)Polycyclic presentation:
2^6,5|b^2=d,d^2=f,c^a=cde,e^a=ef,g^a=g^4,c^b=cef,e^b=ef,d^c=df38 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#65038, #10610434, #10610450, … |
| maximal normal subgroup of | C2×#65038, #10610434, #10610450, … |
| maximal quotient of | #10599188, #10599189, #10599192, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 376);