Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C5×#104 ⋊ C2 |
| Sylow | C5 ⋊ #453 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #4028 |
| commutator, cosocle | C2×C4×C5 | C23 |
| Frattini | C2×C4 | C22×D5 |
| Fitting | C5×#104 | C2 |
| socle | C2×C5 | #120 |
| hypercenter | #104 | D5 |
| hypocenter | C5 | #453 |
| LCS | G ▷ C2×C4×C5 ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #104 | |
168 conjugacy classes of subgroups; only the 39 normal subgroups are shown.
This group has 168 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) = 13 |
| transitive degree | μt(G) = 40 |
A minimal faithful permutation representation — 3 generators on 13 points:
(1,2,4,5,3)(7,10,11,12);(6,7)(8,10)(9,11)(12,13);(2,3)(4,5)(8,13)(10,12)Polycyclic presentation:
2^6,5|c^2=d,e^2=f,c^a=cd,e^a=ef,g^a=g^4,c^b=ce,d^b=df,e^b=ef44 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#65111, #10610717, #10610724, … |
| maximal normal subgroup of | C2×#65111, #10610717, #10610724, … |
| maximal quotient of | #10598979, #10598981, #10598987, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 449);