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.
| Sylow | C5 ⋊ SD64 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #3958 |
| commutator, cosocle | C5×C16 | C22 |
| Frattini | C16 | C2×D5 |
| Fitting, prime radical | C5×D16 | C2 |
| socle | C2×C5 | D16 |
| hypercenter | D16 | D5 |
| hypocenter | C5 | SD64 |
| LCS | G ▷ C5×C16 ▷ C5×C8 ▷ C4×C5 ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ C16 ◁ D16 | |
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)(7,12)(8,17)(9,18)(10,22)(13,14)(15,33)(16,27)(19,36)(20,28)(21,25)(23,31)(24,26)(29,37)(30,32)(34,35);(2,3)(4,5)(6,7,11,16)(8,12,20,27)(9,21,23,37)(10,24,22,15)(13,28,30,17)(14,31,32,18)(19,33,34,26)(25,36,29,35)Polycyclic presentation:
2^6,5|a^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 | #10600169, #10600170, D5×SD64, … |
| maximal normal subgroup of | #10600169, #10600170, D5×SD64, … |
| maximal quotient of | #10598067, #10598069, #10598315, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 78);