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 | C4×C5×D4 | C2 |
| prime radical | C2×C5×D4 | C22 |
| hypercenter | C4×D4 | D5 |
| hypocenter | C5 | #556 |
| LCS | G ▷ C22×C5 ▷ C5 | |
| UCS | C1 ◁ C22 ◁ C4×D4 | |
212 conjugacy classes of subgroups; only the 99 normal subgroups are shown.
This group has 212 conjugacy classes of subgroups — too many to draw. Only its 99 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)(14,17,18,16)(15,19,20,21);(8,12)(10,13)(14,16,18,17)(15,19,20,21);(6,8,9,12)(7,10,11,13);(2,3)(4,5)(6,7,9,11)(8,10,12,13)(14,15,18,20)(16,19,17,21)Polycyclic presentation:
2^6,5|a^2=f,b^2=e,c^2=ef,d^2=ef,d^a=def,g^a=g^4,c^b=ce65 conjugacy classes — view the full character table.
| maximal subgroup of | #10598986, #10598994, #10598998, … |
| maximal normal subgroup of | #10598986, #10598994, #10598998, … |
| maximal quotient of | #10605191, #10605199, #10605208, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 1214);