Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| Fitting | C2×C5×Q16 ⋊ C2 |
| Sylow | C5 ⋊ #497 |
124 conjugacy classes of subgroups; only the 43 normal subgroups are shown.
This group has 124 conjugacy classes of subgroups — too many to draw. Only its 43 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 — 3 generators on 25 points:
(1,2,4,5,3)(7,9)(10,12,14,20)(11,16,17,25)(13,22,21,19)(15,23,24,18);(10,11,13,15,14,17,21,24)(12,18,19,25,20,23,22,16);(2,3)(4,5)(6,7)(8,9)(11,15)(13,21)(16,23)(17,24)(18,25)(19,22)Polycyclic presentation:
2^6,5|b^2=e,c^2=f,e^2=f,b^a=be,c^a=cd,e^a=ef,g^a=g^4,c^b=ce,e^c=ef44 conjugacy classes — view the full character table.
| maximal subgroup of | #10611444, #10611445, #10611457, … |
| maximal normal subgroup of | #10611444, #10611445, #10611457, … |
| maximal quotient of | #10601020, #10601021, #10601043, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 817);