Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| isoclinism | isoclinic class of #64739 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Fitting | C5×Q32 ⋊ C2 |
| Sylow | C5 ⋊ SD64 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #3958 |
| commutator, cosocle | C5×C16 | C22 |
| Frattini | C16 | C2×D5 |
| Fitting | C5×Q32 | C2 |
| socle | C2×C5 | D16 |
| prime radical | D80 | C2 |
| hypercenter | Q32 | D5 |
| hypocenter | C5 | SD64 |
| LCS | G ▷ C5×C16 ▷ C5×C8 ▷ C4×C5 ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ C16 ◁ Q32 | |
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)(6,7,11,15)(8,17,20,34)(9,21,23,37)(10,24,22,14)(12,25,27,16)(13,29,30,18)(19,36,32,28)(26,33,31,35);(2,3)(4,5)(7,12)(8,16)(9,18)(10,22)(13,28)(14,31)(15,27)(17,21)(19,35)(20,25)(23,29)(24,26)(30,36)(32,33)(34,37)Polycyclic presentation:
2^6,5|b^2=f,c^2=de,d^2=ef,e^2=f,b^a=bc,c^a=cd,d^a=de,e^a=ef,g^a=g^4,c^b=cd,d^b=de,e^b=ef41 conjugacy classes — view the full character table.
| maximal subgroup of | D5×SD64, #10600173, #10600176, … |
| maximal normal subgroup of | D5×SD64, #10600173, #10600176, … |
| maximal quotient of | #10598067, #10598068, #10598317, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 79);