Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C5 ⋊ #557 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D5 |
| commutator, socle, prime radical, cosocle | C22×C5 | C24 |
| Fitting | C5×#125 | C2 |
| hypercenter | #125 | D5 |
| hypocenter | C5 | #557 |
| LCS | G ▷ C22×C5 ▷ C5 | |
| UCS | C1 ◁ C22 ◁ #125 | |
200 conjugacy classes of subgroups; only the 105 normal subgroups are shown.
This group has 200 conjugacy classes of subgroups — too many to draw. Only its 105 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) = 320 |
A minimal faithful permutation representation — 4 generators on 21 points:
(1,2,4,5,3)(6,9,10,13)(7,11,8,12)(14,17)(15,19)(16,20)(18,21);(6,8,10,7)(9,11,13,12)(14,16)(15,17)(18,19)(20,21);(14,16,18,19)(15,17,20,21);(2,3)(4,5)(6,7,10,8)(9,12,13,11)(14,15,18,20)(16,17,19,21)Polycyclic presentation:
2^6,5|a^2=f,b^2=e,c^2=ef,d^2=ef,d^a=def,g^a=g^4,c^b=ce,d^c=df53 conjugacy classes — view the full character table.
| maximal subgroup of | #10612882, #10612884, #10612886, … |
| maximal normal subgroup of | #10612882, #10612884, #10612886, … |
| maximal quotient of | #10605888, #10605889, #10605900, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 1357);