Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| Fitting | C5×C4⋊C4 ⋊ C4 |
| Sylow | C5 ⋊ #387 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C22×AGL(1,𝔽5) |
| commutator, socle | C22×C5 | C22×C4 |
| Fitting | C5×C4⋊C4 | C4 |
| prime radical, cosocle | C22×D5 | C23 |
| hypercenter | C4⋊C4 | AGL(1,𝔽5) |
| hypocenter | C5 | #387 |
| LCS | G ▷ C22×C5 ▷ C5 | |
| UCS | C1 ◁ C22 ◁ C4⋊C4 | |
154 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 154 conjugacy classes of subgroups — too many to draw. Only its 54 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)(6,7,8,9)(11,15)(14,17);(7,9)(10,12,13,16)(11,14,15,17);(2,3,4,5)(10,11)(12,14)(13,15)(16,17)Polycyclic presentation:
2^6,5|a^2=d,b^2=e,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 | C2×#65711, #10617101, #10617103, … |
| maximal normal subgroup of | C2×#65711, #10617101, #10617103, … |
| maximal quotient of | #10603788, #10603789, #10603790, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 1049);