Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C5 ⋊ #557 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D5 |
| commutator, socle, prime radical, cosocle | C22×C5 | C24 |
| Fitting | C5×C4⋊Q8 | C2 |
| hypercenter | C4⋊Q8 | D5 |
| hypocenter | C5 | #557 |
| LCS | G ▷ C22×C5 ▷ C5 | |
| UCS | C1 ◁ C22 ◁ C4⋊Q8 | |
200 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 4 generators on 21 points:
(1,2,4,5,3)(6,9,10,8)(7,12,13,11)(14,17,18,21)(15,19,16,20);(8,9)(11,12)(14,16,18,15)(17,19,21,20);(6,8,10,9)(7,11,13,12);(2,3)(4,5)(6,7,10,13)(8,11,9,12)(14,15,18,16)(17,20,21,19)Polycyclic presentation:
2^6,5|a^2=f,b^2=e,c^2=ef,d^2=f,d^a=def,g^a=g^4,c^b=ce,d^c=df53 conjugacy classes — view the full character table.
| maximal subgroup of | #10599249, #10599251, #10599253, … |
| maximal normal subgroup of | #10599249, #10599251, #10599253, … |
| maximal quotient of | #10606054, #10606056, #10606061, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 1393);