Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C2×C5×D8 ⋊ C2 |
| Sylow | C5 ⋊ #447 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | D4×D5 |
| commutator, cosocle | C2×C4×C5 | C23 |
| Frattini | C2×C4 | C22×D5 |
| Fitting | C2×C5×D8 | C2 |
| socle | C22×C5 | C2×D4 |
| hypercenter | C2×D8 | D5 |
| hypocenter | C5 | #447 |
| LCS | G ▷ C2×C4×C5 ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C22 ◁ C2×C4 ◁ C2×D8 | |
198 conjugacy classes of subgroups; only the 45 normal subgroups are shown.
This group has 198 conjugacy classes of subgroups — too many to draw. Only its 45 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:
(1,2,4,5,3)(7,9)(11,14)(12,17)(15,16);(10,11)(12,14)(13,16)(15,17);(2,3)(4,5)(6,7)(8,9)Polycyclic presentation:
2^6,5|e^2=f,c^a=cd,g^a=g^4,c^b=ce,e^b=ef,e^c=ef47 conjugacy classes — view the full character table.
| maximal subgroup of | #10611321, #10611323, #10611325, … |
| maximal normal subgroup of | #10611321, #10611323, #10611325, … |
| maximal quotient of | #10600956, #10600960, #10600991, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 783);