Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| isoclinism | isoclinic class of #64945 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C5 ⋊ #396 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C22×D5 |
| commutator, socle, cosocle | C23×C5 | C23 |
| Fitting | C2×C5×C22⋊C4 | C2 |
| prime radical | C24×C5 | C22 |
| hypercenter | C2×C22⋊C4 | D5 |
| hypocenter | C5 | #396 |
| LCS | G ▷ C23×C5 ▷ C5 | |
| UCS | C1 ◁ C23 ◁ C2×C22⋊C4 | |
186 conjugacy classes of subgroups; only the 63 normal subgroups are shown.
This group has 186 conjugacy classes of subgroups — too many to draw. Only its 63 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 3 generators on 17 points:
(1,2,4,5,3)(7,9)(11,13)(14,15)(16,17);(10,11,12,13)(15,17);(2,3)(4,5)(6,7)(8,9)(14,15,16,17)Polycyclic presentation:
2^6,5|a^2=e,b^2=ef,b^a=be,c^a=cd,g^a=g^4,c^b=cf62 conjugacy classes — view the full character table.
| maximal subgroup of | #10605211, #10605213, #10605219, … |
| maximal normal subgroup of | #10605211, #10605213, #10605219, … |
| maximal quotient of | #10600202, #10600203, #10600204, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 575);