Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| automorphism | ? |
| 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, prime radical | C5×#41 | C4 |
| cosocle | #168 | C23 |
| hypercenter | #41 | AGL(1,𝔽5) |
| hypocenter | C5 | #504 |
| LCS | G ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C4 ◁ #41 | |
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)(8,16)(12,22)(17,28)(18,29)(23,32)(24,33)(30,36)(34,37);(6,8)(7,12)(9,17)(10,18)(11,16)(13,23)(14,24)(15,22)(19,30)(20,28)(21,29)(25,34)(26,32)(27,33)(31,36)(35,37);(2,3,4,5)(6,7,9,13,10,14,19,25,11,15,20,26,21,27,31,35)(8,12,17,23,18,24,30,34,16,22,28,32,29,33,36,37)Polycyclic presentation:
2^6,5|a^2=d,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, #10604861, … |
| maximal normal subgroup of | #10603999, #10604000, #10604861, … |
| maximal quotient of | #10604366, #10604367, #10604368, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 1129);