Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| Fitting | C5×#102 ⋊ C2 |
| Sylow | C5 ⋊ #469 |
| 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 | #469 |
| LCS | G ▷ C2×C4×C5 ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C22 ◁ C2×C4 ◁ #102 | |
130 conjugacy classes of subgroups; only the 39 normal subgroups are shown.
This group has 130 conjugacy classes of subgroups — too many to draw. Only its 39 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)(12,17)(14,16);(6,7,8,9)(10,11,13,14)(12,15,17,16);(2,3)(4,5)(7,9)(11,14)(15,16)Polycyclic presentation:
2^6,5|b^2=e,d^2=f,b^a=be,c^a=cef,g^a=g^4,c^b=cd,d^b=df,d^c=df44 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#65071, #10610483, #10610492, … |
| maximal normal subgroup of | C2×#65071, #10610483, #10610492, … |
| maximal quotient of | #10599192, #10599194, #10599206, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 409);