Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C5×#131 ⋊ C2 |
| Sylow | C5 ⋊ #576 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C22×D20 |
| commutator, cosocle | C4×C5 | C24 |
| Frattini | C4 | C23×D5 |
| Fitting | C5×#131 | C2 |
| socle | C2×C5 | C22×D4 |
| hypercenter | #131 | D5 |
| hypocenter | C5 | #576 |
| LCS | G ▷ C4×C5 ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C2 ◁ #41 ◁ #131 | |
268 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 80 |
A minimal faithful permutation representation — 4 generators on 21 points:
(1,2,4,5,3)(6,8,15,20,10,16,9,14)(7,12,18,21,11,17,13,19);(7,11)(12,17)(13,18)(19,21);(6,7)(8,12)(9,13)(10,11)(14,19)(15,18)(16,17)(20,21);(2,3)(4,5)(8,14)(9,15)(12,19)(13,18)(16,20)(17,21)Polycyclic presentation:
2^6,5|d^2=ef,e^2=f,d^a=de,e^a=ef,g^a=g^4,c^b=cf62 conjugacy classes — view the full character table.
| maximal subgroup of | #10599919, #10599921, #10601887, … |
| maximal normal subgroup of | #10599919, #10599921, #10601887, … |
| maximal quotient of | #10610277, #10610291, #10610292, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 1424);