Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C5×#121 ⋊ C2 |
| Sylow | C5 ⋊ #553 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D5 |
| commutator, socle, cosocle | C22×C5 | C24 |
| Fitting | C5×#121 | C2 |
| hypercenter | #121 | D5 |
| hypocenter | C5 | #553 |
| LCS | G ▷ C22×C5 ▷ C5 | |
| UCS | C1 ◁ C22 ◁ #121 | |
238 conjugacy classes of subgroups; only the 97 normal subgroups are shown.
This group has 238 conjugacy classes of subgroups — too many to draw. Only its 97 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) = 160 |
A minimal faithful permutation representation — 4 generators on 21 points:
(1,2,4,5,3)(8,13)(10,12)(14,16)(15,19)(17,20)(18,21);(6,8,9,13)(7,10,11,12)(15,18)(16,20);(16,20)(19,21);(2,3)(4,5)(6,7,9,11)(8,12,13,10)(14,15,17,18)(16,19,20,21)Polycyclic presentation:
2^6,5|a^2=f,c^2=e,c^a=cf,g^a=g^4,d^b=def,d^c=df53 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#65935, #10617838, #10617852, … |
| maximal normal subgroup of | C2×#65935, #10617838, #10617852, … |
| maximal quotient of | #10605611, #10605615, #10605633, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 1273);