Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C5×#124 ⋊ C2 |
| Sylow | C5 ⋊ #454 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #4083 |
| commutator, cosocle | C2×C4×C5 | C23 |
| Frattini | C2×C4 | C22×D5 |
| Fitting | C5×#124 | C2 |
| socle | C2×C5 | #120 |
| hypercenter | #124 | D5 |
| hypocenter | C5 | #454 |
| LCS | G ▷ C2×C4×C5 ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #124 | |
146 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)(7,12)(9,18)(11,19)(13,14)(15,17)(20,21);(6,8,10,16)(7,11,14,21)(9,17,18,15)(12,20,13,19);(2,3)(4,5)(6,7)(8,11)(9,12)(10,14)(13,18)(15,19)(16,21)(17,20)Polycyclic presentation:
2^6,5|b^2=f,d^2=f,c^a=cd,d^a=df,e^a=ef,g^a=g^4,c^b=ce,d^b=df,d^c=df44 conjugacy classes — view the full character table.
| maximal subgroup of | #10597951, #10598033, #10599953, … |
| maximal normal subgroup of | #10597951, #10598033, #10599953, … |
| maximal quotient of | #10600336, #10600355, #10600356, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 689);