Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C4×C5×D4 ⋊ C2 |
| Sylow | C5 ⋊ #547 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D5 |
| commutator, socle, cosocle | C22×C5 | C24 |
| Fitting | C4×C5×D4 | C2 |
| hypercenter | C4×D4 | D5 |
| hypocenter | C5 | #547 |
| LCS | G ▷ C22×C5 ▷ C5 | |
| UCS | C1 ◁ C22 ◁ C4×D4 | |
292 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 292 conjugacy classes of subgroups — too many to draw. Only its 115 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 4 generators on 17 points:
(1,2,4,5,3)(6,7,8,9);(11,15)(12,16);(10,12,13,16)(11,14,15,17);(2,3)(4,5)(7,9)(10,11)(12,14)(13,15)(16,17)Polycyclic presentation:
2^6,5|b^2=f,d^2=e,c^a=cf,d^a=de,g^a=g^4,c^b=cf65 conjugacy classes — view the full character table.
| maximal subgroup of | #10598978, #10598993, #10598997, … |
| maximal normal subgroup of | #10598978, #10598993, #10598997, … |
| maximal quotient of | #10605187, #10605195, #10605198, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 1227);