Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C5×#131 ⋊ C2 |
| Sylow | C5 ⋊ #472 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #4076 |
| commutator, cosocle | C2×C4×C5 | C23 |
| Frattini | C2×C4 | C22×D5 |
| Fitting | C5×#131 | C2 |
| socle | C2×C5 | #121 |
| hypercenter | #131 | D5 |
| hypocenter | C5 | #472 |
| LCS | G ▷ C2×C4×C5 ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #131 | |
108 conjugacy classes of subgroups.
| 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)(6,8)(7,13)(9,16)(10,17)(11,18)(12,15)(14,21)(19,20);(6,7,12,19,10,14,9,11)(8,13,15,20,17,21,16,18);(2,3)(4,5)(7,11)(8,15)(9,12)(14,19)(16,17)(18,20)Polycyclic presentation:
2^6,5|b^2=ef,d^2=f,e^2=f,b^a=be,c^a=cd,d^a=df,e^a=ef,g^a=g^4,d^c=df56 conjugacy classes — view the full character table.
| maximal subgroup of | #10599919, #10599920, #10601885, … |
| maximal normal subgroup of | #10599919, #10599920, #10601885, … |
| maximal quotient of | #10600685, #10600687, #10600696, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 769);