Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C5×#102 ⋊ C2 |
| Sylow | C5 ⋊ #466 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | D4×D5 |
| commutator, cosocle | C2×C4×C5 | C23 |
| Frattini | C2×C4 | C22×D5 |
| Fitting | C5×#102 | C2 |
| socle | C22×C5 | C2×D4 |
| hypercenter | #102 | D5 |
| hypocenter | C5 | #466 |
| LCS | G ▷ C2×C4×C5 ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C22 ◁ C2×C4 ◁ #102 | |
134 conjugacy classes of subgroups; only the 41 normal subgroups are shown.
This group has 134 conjugacy classes of subgroups — too many to draw. Only its 41 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) = 160 |
A minimal faithful permutation representation — 3 generators on 17 points:
(1,2,4,5,3)(6,7)(8,9)(11,15)(12,13)(16,17);(7,9)(10,11,12,16,14,17,13,15);(2,3)(4,5)(7,9)Polycyclic presentation:
2^6,5|b^2=d,d^2=f,e^2=f,c^a=cde,g^a=g^4,c^b=cef,d^c=df,e^c=ef50 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#65064, #10610483, #10610488, … |
| maximal normal subgroup of | C2×#65064, #10610483, #10610488, … |
| maximal quotient of | #10599188, #10599190, #10599202, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 402);