Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| automorphism | Aut(G) = C4×D4×AGL(1,𝔽5) (order 640) · Inn(G) = C22×AGL(1,𝔽5) · Out(G) = C2×C4 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C5 ⋊ #504 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C4 | C22×AGL(1,𝔽5) |
| commutator, socle | C2×C5 | C22×C8 |
| Fitting | C5×M16 | C4 |
| prime radical | #709 | C4 |
| cosocle | #168 | C23 |
| hypercenter | M16 | AGL(1,𝔽5) |
| hypocenter | C5 | #504 |
| LCS | G ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C4 ◁ M16 | |
84 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 3 generators on 37 points:
(1,2,3,5,4)(6,9,11,23,12,10,22,24)(7,14,16,30,17,15,29,31)(8,18,19,32,20,21,33,34)(13,25,26,35,27,28,36,37);(6,8)(7,13)(9,21)(10,18)(11,19)(12,20)(14,28)(15,25)(16,26)(17,27)(22,33)(23,34)(24,32)(29,36)(30,37)(31,35);(2,3,4,5)(6,7,10,15,11,16,24,31,12,17,9,14,22,29,23,30)(8,13,18,25,19,26,32,35,20,27,21,28,33,36,34,37)Polycyclic presentation:
2^6,5|a^2=d,c^2=e,d^2=e,e^2=f,g^a=g^2,c^b=cf,g^d=g^450 conjugacy classes — view the full character table.
| maximal subgroup of | #10603999, #10604000, #10604083, … |
| maximal normal subgroup of | #10603999, #10604000, #10604083, … |
| maximal quotient of | #10603868, #10603869, #10603870, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 1063);