Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
| unambiguous semidirect product | (C2×AGL(1,𝔽5))⋊Q8 |
| Fitting | C2×C5×Q8 ⋊ C4 |
| Sylow | C5 ⋊ #391 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C22×AGL(1,𝔽5) |
| commutator, socle | C22×C5 | C22×C4 |
| Fitting | C2×C5×Q8 | C4 |
| prime radical, cosocle | C22×D5 | C23 |
| hypercenter | C2×Q8 | AGL(1,𝔽5) |
| hypocenter | C5 | #391 |
| LCS | G ▷ C22×C5 ▷ C5 | |
| UCS | C1 ◁ C22 ◁ C2×Q8 | |
186 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 186 conjugacy classes of subgroups — too many to draw. Only its 64 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 80 |
A minimal faithful permutation representation — 3 generators on 17 points:
(1,2,3,5,4)(7,9)(10,12,13,17)(11,14,15,16);(10,11,13,15)(12,16,17,14);(2,3,4,5)(6,7)(8,9)Polycyclic presentation:
2^6,5|a^2=d,b^2=f,c^2=f,c^a=ce,g^a=g^2,c^b=cf,g^d=g^438 conjugacy classes — view the full character table.
| maximal subgroup of | #10598577, #10603931, #10603985, … |
| maximal normal subgroup of | #10598577, #10603931, #10603985, … |
| maximal quotient of | #10604310, #10604311, #10604330, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 1128);