Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C5×#41 ⋊ C4 |
| Sylow | C5 ⋊ #520 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C23×AGL(1,𝔽5) |
| commutator, socle | C2×C5 | C23×C4 |
| Fitting | C5×#41 | C4 |
| prime radical | D5×#41 | C2 |
| cosocle | C2×D5 | C24 |
| hypercenter | #41 | AGL(1,𝔽5) |
| hypocenter | C5 | #520 |
| LCS | G ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C2 ◁ #41 | |
294 conjugacy classes of subgroups; only the 138 normal subgroups are shown.
This group has 294 conjugacy classes of subgroups — too many to draw. Only its 138 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 13 |
| transitive degree | μt(G) = 40 |
A minimal faithful permutation representation — 4 generators on 13 points:
(1,2,3,5,4)(6,8,9,12)(7,11,10,13);(7,10)(11,13);(6,7)(8,11)(9,10)(12,13);(2,3,4,5)(8,12)(11,13)Polycyclic presentation:
2^6,5|a^2=e,d^2=f,d^a=df,g^a=g^2,c^b=cf,g^e=g^444 conjugacy classes — view the full character table.
| maximal subgroup of | #10604003, #10604004, #10604863, … |
| maximal normal subgroup of | #10604003, #10604004, #10604863, … |
| maximal quotient of | #10617120, #10617121, #10617122, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 1604);