Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| isoclinism | isoclinic class of D160 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Fitting | C5×C32 ⋊ C2 |
| Sylow | C5 ⋊ SD64 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | D80 |
| commutator, cosocle | C5×C16 | C22 |
| Frattini | C16 | C2×D5 |
| Fitting | C5×C32 | C2 |
| socle | C2×C5 | D16 |
| prime radical | D80 | C2 |
| hypercenter | C32 | D5 |
| hypocenter | C5 | SD64 |
| LCS | G ▷ C5×C16 ▷ C5×C8 ▷ C4×C5 ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ C16 ◁ C32 | |
32 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 2 generators on 37 points:
(1,2,4,5,3)(6,7,13,25,22,30,34,31,10,15,28,37,21,36,8,12,11,16,17,33,9,14,19,26,23,32,35,29,18,24,20,27);(2,3)(4,5)(7,12)(8,17)(9,21)(10,23)(13,20)(14,29)(15,31)(16,27)(18,22)(19,28)(24,33)(25,36)(26,32)(30,37)(34,35)Polycyclic presentation:
2^6,5|b^2=cd,c^2=de,d^2=e,e^2=f,b^a=bc,c^a=cdf,d^a=def,e^a=ef,g^a=g^483 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#64669, #10600162, #10600166, … |
| maximal normal subgroup of | C2×#64669, #10600162, #10600166, … |
| maximal quotient of | #10598083, #10598087, #10598088, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 7);