Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C5 ⋊ #556 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D5 |
| commutator, socle, cosocle | C22×C5 | C24 |
| Fitting | C5×#124 | C2 |
| prime radical | C2×C5×D4 | C22 |
| hypercenter | #124 | D5 |
| hypocenter | C5 | #556 |
| LCS | G ▷ C22×C5 ▷ C5 | |
| UCS | C1 ◁ C22 ◁ #124 | |
212 conjugacy classes of subgroups; only the 97 normal subgroups are shown.
This group has 212 conjugacy classes of subgroups — too many to draw. Only its 97 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) = 160 |
A minimal faithful permutation representation — 4 generators on 21 points:
(1,2,4,5,3)(8,13)(10,12)(15,19)(18,21);(6,8,9,13)(7,10,11,12)(14,16,17,20)(15,18,19,21);(8,13)(10,12)(14,15)(16,18)(17,19)(20,21);(2,3)(4,5)(6,7,9,11)(8,12,13,10)Polycyclic presentation:
2^6,5|a^2=f,c^2=e,c^a=cf,g^a=g^4,c^b=cf,d^b=def,d^c=df53 conjugacy classes — view the full character table.
| maximal subgroup of | #10612247, #10612249, #10612253, … |
| maximal normal subgroup of | #10612247, #10612249, #10612253, … |
| maximal quotient of | #10605889, #10605890, #10605904, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 1343);