Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C2×C5×D8 ⋊ C2 |
| Sylow | C5 ⋊ #472 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #4084 |
| commutator, cosocle | C2×C4×C5 | C23 |
| Frattini | C2×C4 | C22×D5 |
| Fitting | C2×C5×D8 | C2 |
| socle | C2×C5 | #121 |
| hypercenter | C2×D8 | D5 |
| hypocenter | C5 | #472 |
| LCS | G ▷ C2×C4×C5 ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×D8 | |
108 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 80 |
A minimal faithful permutation representation — 3 generators on 21 points:
(1,2,4,5,3)(7,13)(8,16)(9,10)(12,19)(14,15)(17,18);(6,8,10,17,11,18,9,16)(7,12,14,20,15,19,13,21);(2,3)(4,5)(6,7)(8,12)(9,13)(10,14)(11,15)(16,21)(17,20)(18,19)Polycyclic presentation:
2^6,5|b^2=e,d^2=f,e^2=f,c^a=cd,d^a=df,g^a=g^4,c^b=ce,d^c=df,e^c=ef44 conjugacy classes — view the full character table.
| maximal subgroup of | #10600053, #10600054, #10602264, … |
| maximal normal subgroup of | #10600053, #10600054, #10602264, … |
| maximal quotient of | #10600988, #10600989, #10600990, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 787);