Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C2×C5×M16 ⋊ C2 |
| Sylow | C5 ⋊ #568 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C4 | C23×D5 |
| commutator, socle | C2×C5 | C23×C4 |
| Fitting | C2×C5×M16 | C2 |
| prime radical | C2×#709 | C2 |
| cosocle | C4×C5 | C24 |
| hypercenter | C2×M16 | D5 |
| hypocenter | C5 | #568 |
| LCS | G ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C4 ◁ C2×M16 | |
258 conjugacy classes of subgroups; only the 147 normal subgroups are shown.
This group has 258 conjugacy classes of subgroups — too many to draw. Only its 147 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 80 |
A minimal faithful permutation representation — 4 generators on 21 points:
(1,2,4,5,3)(6,8,9,15,10,14,16,20)(7,12,13,19,11,17,18,21);(7,11)(12,17)(13,18)(19,21);(8,14)(12,17)(15,20)(19,21);(2,3)(4,5)(6,7)(8,12)(9,13)(10,11)(14,17)(15,19)(16,18)(20,21)Polycyclic presentation:
2^6,5|d^2=e,e^2=f,c^a=cf,g^a=g^4,d^b=df74 conjugacy classes — view the full character table.
| maximal subgroup of | #10599505, #10599506, #10599518, … |
| maximal normal subgroup of | #10599505, #10599506, #10599518, … |
| maximal quotient of | #10610211, #10610212, #10610213, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 1417);