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 ⋊ Q64 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #3958 |
| commutator, cosocle | C5×C16 | C22 |
| Frattini | C16 | C2×D5 |
| Fitting | C5×Q32 | C2 |
| socle, prime radical | C2×C5 | D16 |
| hypercenter | Q32 | D5 |
| hypocenter | C5 | Q64 |
| LCS | G ▷ C5×C16 ▷ C5×C8 ▷ C4×C5 ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ C16 ◁ Q32 | |
30 conjugacy classes of subgroups.
| minimal degree | μ(G) = 69 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 2 generators on 69 points:
(1,2,4,5,3)(6,8,12,22)(7,13,17,38)(9,24,28,58)(10,30,32,64)(11,34,31,21)(14,40,42,67)(15,44,45,69)(16,46,33,37)(18,47,49,23)(19,39,35,26)(20,43,52,25)(27,62,60,57)(29,36,63,65)(41,68,54,51)(48,55,53,61)(50,59,56,66);(2,3)(4,5)(6,7,12,17)(8,18,22,49)(9,23,28,47)(10,29,32,63)(11,33,31,16)(13,35,38,19)(14,39,42,26)(15,43,45,25)(20,51,52,68)(21,53,34,48)(24,44,58,69)(27,61,60,55)(30,40,64,67)(36,62,65,57)(37,50,46,56)(41,59,54,66)Polycyclic presentation:
2^6,5|a^2=f,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 | #10600172, #10600174, D5×Q64, … |
| maximal normal subgroup of | #10600172, #10600174, D5×Q64, … |
| maximal quotient of | #10598066, #10598069, #10598317, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 80);