Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C2×C5×M16 ⋊ C2 |
| Sylow | C5 ⋊ #407 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2×C4 | C22×D5 |
| commutator, socle | C22×C5 | C22×C4 |
| Fitting | C2×C5×M16 | C2 |
| prime radical | C23×D5 | C4 |
| cosocle | C2×C4×C5 | C23 |
| hypercenter | C2×M16 | D5 |
| hypocenter | C5 | #407 |
| LCS | G ▷ C22×C5 ▷ C5 | |
| UCS | C1 ◁ C2×C4 ◁ C2×M16 | |
190 conjugacy classes of subgroups.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 80 |
A minimal faithful permutation representation — 3 generators on 17 points:
(1,2,4,5,3)(7,9)(10,11,12,15,13,14,16,17);(11,14)(15,17);(2,3)(4,5)(6,7)(8,9)Polycyclic presentation:
2^6,5|c^2=d,d^2=f,c^a=ce,g^a=g^4,c^b=cf68 conjugacy classes — view the full character table.
| maximal subgroup of | #10598021, #10598783, #10598785, … |
| maximal normal subgroup of | #10598021, #10598783, #10598785, … |
| maximal quotient of | #10600600, #10600630, #10600631, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 753);