Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| Fitting | C5×#124 ⋊ C2 |
| Sylow | C5 ⋊ H(3,𝔽4) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D5 |
| commutator, socle, cosocle | C22×C5 | C24 |
| Fitting | C5×#124 | C2 |
| hypercenter | #124 | D5 |
| hypocenter | C5 | H(3,𝔽4) |
| LCS | G ▷ C22×C5 ▷ C5 | |
| UCS | C1 ◁ C22 ◁ #124 | |
260 conjugacy classes of subgroups; only the 91 normal subgroups are shown.
This group has 260 conjugacy classes of subgroups — too many to draw. Only its 91 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)(6,8,9,13)(7,11,10,12)(15,19)(16,20);(7,10)(8,13)(14,16)(15,18)(17,20)(19,21);(8,13)(11,12)(14,15)(16,21)(17,19)(18,20);(2,3)(4,5)(6,7)(8,12)(9,10)(11,13)(16,20)(18,21)Polycyclic presentation:
2^6,5|d^2=ef,c^a=cf,d^a=def,g^a=g^4,c^b=ce,d^b=df,d^c=df47 conjugacy classes — view the full character table.
| maximal subgroup of | #10599143, #10599145, #10599146, … |
| maximal normal subgroup of | #10599143, #10599145, #10599146, … |
| maximal quotient of | #10605935, #10605946, #10605949, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 1355);