Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C5×2−1+4 ⋊ C2 |
| Sylow | C5 ⋊ #454 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #4099 |
| commutator, cosocle | C2×C4×C5 | C23 |
| Frattini | C2×C4 | C22×D5 |
| Fitting | C5×2−1+4 | C2 |
| socle | C2×C5 | #120 |
| hypercenter | 2−1+4 | D5 |
| hypocenter | C5 | #454 |
| LCS | G ▷ C2×C4×C5 ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ 2−1+4 | |
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)(6,8,10,15)(7,12,13,21)(9,18,16,14)(11,19,17,20);(6,7)(8,12)(9,17)(10,13)(11,16)(14,19)(15,21)(18,20);(2,3)(4,5)(7,11)(9,16)(12,20)(13,17)(14,18)(19,21)Polycyclic presentation:
2^6,5|c^2=f,e^2=f,b^a=be,c^a=cd,e^a=ef,g^a=g^4,d^b=df,e^b=ef,e^c=ef50 conjugacy classes — view the full character table.
| maximal subgroup of | #10598835, #10601780, #10601781, … |
| maximal normal subgroup of | #10598835, #10601780, #10601781, … |
| maximal quotient of | #10602451, #10602454, #10602458, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 872);