Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C4×C5×Q8 ⋊ C2 |
| Sylow | C5 ⋊ #556 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D5 |
| commutator, socle, cosocle | C22×C5 | C24 |
| Fitting | C4×C5×Q8 | C2 |
| prime radical | C2×D20 | C22 |
| hypercenter | C4×Q8 | D5 |
| hypocenter | C5 | #556 |
| LCS | G ▷ C22×C5 ▷ C5 | |
| UCS | C1 ◁ C22 ◁ C4×Q8 | |
212 conjugacy classes of subgroups; only the 97 normal subgroups are shown.
This group has 212 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)(6,9,10,8)(7,12,13,11)(15,19)(18,21);(6,8,10,9)(7,12,13,11)(14,16,17,20)(15,18,19,21);(6,8,10,9)(7,11,13,12)(14,15)(16,18)(17,19)(20,21);(2,3)(4,5)(6,7,10,13)(8,11,9,12)Polycyclic presentation:
2^6,5|a^2=f,b^2=f,c^2=e,d^2=f,c^a=cf,g^a=g^4,d^b=def65 conjugacy classes — view the full character table.
| maximal subgroup of | #10611351, #10611361, #10611370, … |
| maximal normal subgroup of | #10611351, #10611361, #10611370, … |
| maximal quotient of | #10605366, #10605370, #10605373, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 1240);