Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C5 ⋊ #544 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D5 |
| commutator, socle, cosocle | C22×C5 | C24 |
| Fitting | C2×C5×C4⋊C4 | C2 |
| prime radical | C23×C5 | C23 |
| hypercenter | C2×C4⋊C4 | D5 |
| hypocenter | C5 | #544 |
| LCS | G ▷ C22×C5 ▷ C5 | |
| UCS | C1 ◁ C22 ◁ C2×C4⋊C4 | |
206 conjugacy classes of subgroups.
| 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)(8,12)(10,13)(14,16,17,21)(15,18,19,20);(7,11)(10,13);(6,8,9,12)(7,10,11,13);(2,3)(4,5)(6,7)(8,10)(9,11)(12,13)(14,15,17,19)(16,20,21,18)Polycyclic presentation:
2^6,5|a^2=e,b^2=f,d^2=e,c^a=cf,d^a=de,g^a=g^4,d^b=df62 conjugacy classes — view the full character table.
| maximal subgroup of | #10598931, #10598932, #10598933, … |
| maximal normal subgroup of | #10598931, #10598932, #10598933, … |
| maximal quotient of | #10605065, #10605081, #10605091, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 1172);