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 ⋊ #554 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D5 |
| commutator, socle, cosocle | C22×C5 | C24 |
| Fitting | C5×C22⋊Q8 | C2 |
| prime radical | C2×C22⋊D5 | C22 |
| hypercenter | C22⋊Q8 | D5 |
| hypocenter | C5 | #554 |
| LCS | G ▷ C22×C5 ▷ C5 | |
| UCS | C1 ◁ C22 ◁ C22⋊Q8 | |
228 conjugacy classes of subgroups; only the 105 normal subgroups are shown.
This group has 228 conjugacy classes of subgroups — too many to draw. Only its 105 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)(6,8)(7,11)(9,12)(10,13)(14,17,18,16)(15,20,19,21);(8,12)(11,13)(14,16,18,17)(15,20,19,21);(7,10)(8,12)(14,15)(16,20)(17,21)(18,19);(2,3)(4,5)(6,7)(8,11)(9,10)(12,13)(15,19)(20,21)Polycyclic presentation:
2^6,5|c^2=e,d^2=e,b^a=bf,g^a=g^4,d^b=df,d^c=def53 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#65970, #10617887, #10617888, … |
| maximal normal subgroup of | C2×#65970, #10617887, #10617888, … |
| maximal quotient of | #10605611, #10605620, #10605634, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 1308);