Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C5 ⋊ Q82 |
| 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 | Q82 |
| LCS | G ▷ C22×C5 ▷ C5 | |
| UCS | C1 ◁ C22 ◁ C4⋊Q8 | |
212 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)(6,8)(7,11)(9,13)(10,12)(14,16,17,21)(15,18,19,20);(6,8,9,13)(7,11,12,10)(14,15,17,19)(16,20,21,18);(6,8,9,13)(7,10,12,11);(2,3)(4,5)(6,7,9,12)(8,11,13,10)Polycyclic presentation:
2^6,5|a^2=f,b^2=f,c^2=e,d^2=ef,b^a=bf,g^a=g^4,d^b=df,d^c=de53 conjugacy classes — view the full character table.
| maximal subgroup of | #10599297, #10599298, #10599300, … |
| maximal normal subgroup of | #10599297, #10599298, #10599300, … |
| maximal quotient of | #10606054, #10606062, #10606066, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 1394);