Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C5×#117 ⋊ C2 |
| Sylow | C5 ⋊ C4×#41 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D5 |
| commutator | C2×C5 | C23×C4 |
| Frattini | C22 | C23×D5 |
| Fitting | C5×#117 | C2 |
| socle, cosocle | C22×C5 | C24 |
| prime radical | C2×#709 | C2 |
| hypercenter | #117 | D5 |
| hypocenter | C5 | C4×#41 |
| LCS | G ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C2×C4 ◁ #117 | |
310 conjugacy classes of subgroups; only the 155 normal subgroups are shown.
This group has 310 conjugacy classes of subgroups — too many to draw. Only its 155 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 — 4 generators on 17 points:
(1,2,4,5,3)(6,7,8,9)(12,16)(14,17);(10,12)(11,14)(13,16)(15,17);(10,11,13,15)(12,14,16,17);(2,3)(4,5)(12,16)(14,17)Polycyclic presentation:
2^6,5|b^2=f,d^2=e,c^a=cf,g^a=g^4,d^c=df80 conjugacy classes — view the full character table.
| maximal subgroup of | #10599504, #10599521, #10599522, … |
| maximal normal subgroup of | #10599504, #10599521, #10599522, … |
| maximal quotient of | #10605113, #10605115, #10605116, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 1194);