Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C5×#117 ⋊ C2 |
| Sylow | C5 ⋊ #536 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D5 |
| commutator, socle, cosocle | C22×C5 | C24 |
| Fitting | C5×#117 | C2 |
| prime radical | C2×#709 | C2 |
| hypercenter | #117 | D5 |
| hypocenter | C5 | #536 |
| LCS | G ▷ C22×C5 ▷ C5 | |
| UCS | C1 ◁ C22 ◁ #117 | |
270 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 270 conjugacy classes of subgroups — too many to draw. Only its 111 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 4 generators on 25 points:
(1,2,4,5,3)(6,7,8,9)(11,17)(12,19)(13,21)(22,24);(10,13)(11,16)(12,18)(14,21)(15,22)(17,20)(19,25)(23,24);(10,12,14,19)(11,15,17,23)(13,18,21,25)(16,22,20,24);(2,3)(4,5)(7,9)(10,11)(12,15)(13,20)(14,17)(16,21)(18,24)(19,23)(22,25)Polycyclic presentation:
2^6,5|b^2=f,d^2=ef,c^a=cf,d^a=de,g^a=g^4,d^b=df,d^c=df62 conjugacy classes — view the full character table.
| maximal subgroup of | #10599501, #10599505, #10599527, … |
| maximal normal subgroup of | #10599501, #10599505, #10599527, … |
| maximal quotient of | #10605118, #10605119, #10605134, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 1198);