Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C5×C22⋊Q8 ⋊ C2 |
| Sylow | C5 ⋊ #538 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D5 |
| commutator, socle, cosocle | C22×C5 | C24 |
| Fitting | C5×C22⋊Q8 | C2 |
| hypercenter | C22⋊Q8 | D5 |
| hypocenter | C5 | #538 |
| LCS | G ▷ C22×C5 ▷ C5 | |
| UCS | C1 ◁ C22 ◁ C22⋊Q8 | |
250 conjugacy classes of subgroups; only the 95 normal subgroups are shown.
This group has 250 conjugacy classes of subgroups — too many to draw. Only its 95 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)(14,16,17,21)(15,19,18,20);(8,12)(10,13)(14,15,17,18)(16,20,21,19);(6,8)(7,10)(9,12)(11,13);(2,3)(4,5)(6,7)(8,10)(9,11)(12,13)(15,18)(19,20)Polycyclic presentation:
2^6,5|c^2=f,d^2=f,c^a=cf,d^a=de,g^a=g^4,c^b=ce,d^c=df50 conjugacy classes — view the full character table.
| maximal subgroup of | #10599110, #10599112, #10599114, … |
| maximal normal subgroup of | #10599110, #10599112, #10599114, … |
| maximal quotient of | #10605604, #10605659, #10605687, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 1316);