Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C5 ⋊ #446 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C22×AGL(1,𝔽5) |
| commutator, socle, prime radical | C22×C5 | C22×C4 |
| Fitting | C5×C4⋊C4 | C4 |
| cosocle | C2×Dic5 | C23 |
| hypercenter | C4⋊C4 | AGL(1,𝔽5) |
| hypocenter | C5 | #446 |
| LCS | G ▷ C22×C5 ▷ C5 | |
| UCS | C1 ◁ C22 ◁ C4⋊C4 | |
94 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 3 generators on 21 points:
(1,2,3,5,4)(6,8,9,12)(7,10,11,13)(15,20)(19,21);(6,7,9,11)(8,13,12,10)(14,16,18,17)(15,19,20,21);(2,3,4,5)(8,12)(10,13)(14,15,17,21,18,20,16,19)Polycyclic presentation:
2^6,5|a^2=d,b^2=ef,c^2=e,d^2=f,c^a=cef,g^a=g^2,c^b=ce,g^d=g^438 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#65707, #10617088, #10617094, … |
| maximal normal subgroup of | C2×#65707, #10617088, #10617094, … |
| maximal quotient of | #10603780, #10603781, #10603788, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 1045);