Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C5×#123 ⋊ C2 |
| Sylow | C5 ⋊ #538 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D5 |
| commutator, socle, cosocle | C22×C5 | C24 |
| Fitting | C5×#123 | C2 |
| hypercenter | #123 | D5 |
| hypocenter | C5 | #538 |
| LCS | G ▷ C22×C5 ▷ C5 | |
| UCS | C1 ◁ C22 ◁ #123 | |
250 conjugacy classes of subgroups; only the 93 normal subgroups are shown.
This group has 250 conjugacy classes of subgroups — too many to draw. Only its 93 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)(7,11)(10,13)(15,19)(18,21);(8,12)(10,13)(14,16,17,20)(15,18,19,21);(6,8)(7,10)(9,12)(11,13)(14,15)(16,18)(17,19)(20,21);(2,3)(4,5)(6,7)(8,10)(9,11)(12,13)Polycyclic presentation:
2^6,5|c^2=f,d^a=de,g^a=g^4,c^b=ce,d^b=df50 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#65991, #10617914, #10617917, … |
| maximal normal subgroup of | C2×#65991, #10617914, #10617917, … |
| maximal quotient of | #10605789, #10605801, #10605815, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 1329);