Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C5×#108 ⋊ C2 |
| Sylow | C5 ⋊ #472 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #4041 |
| commutator, cosocle | C2×C4×C5 | C23 |
| Frattini | C2×C4 | C22×D5 |
| Fitting | C5×#108 | C2 |
| socle | C2×C5 | #121 |
| hypercenter | #108 | D5 |
| hypocenter | C5 | #472 |
| LCS | G ▷ C2×C4×C5 ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #108 | |
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)(6,9,11,15,12,14,10,7)(8,16,18,21,19,13,17,20);(6,8,11,18,12,19,10,17)(7,13,14,21,15,16,9,20);(2,3)(4,5)(6,7)(8,13)(9,10)(11,14)(12,15)(16,19)(17,20)(18,21)Polycyclic presentation:
2^6,5|b^2=e,c^2=df,d^2=f,e^2=f,c^a=cd,d^a=df,g^a=g^4,c^b=ce,e^c=ef44 conjugacy classes — view the full character table.
| maximal subgroup of | #10600035, #10600036, #10600051, … |
| maximal normal subgroup of | #10600035, #10600036, #10600051, … |
| maximal quotient of | #10599331, #10599333, #10599336, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 524);