Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C5 ⋊ #557 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D5 |
| commutator, socle, prime radical, cosocle | C22×C5 | C24 |
| Fitting | C4×C5×Q8 | C2 |
| hypercenter | C4×Q8 | D5 |
| hypocenter | C5 | #557 |
| LCS | G ▷ C22×C5 ▷ C5 | |
| UCS | C1 ◁ C22 ◁ C4×Q8 | |
200 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 4 generators on 21 points:
(1,2,4,5,3)(6,8,9,13)(7,10,11,12);(14,17,18,15)(16,19,20,21);(14,16,18,20)(15,19,17,21);(2,3)(4,5)(6,7,9,11)(8,12,13,10)(14,15,18,17)(16,19,20,21)Polycyclic presentation:
2^6,5|a^2=e,b^2=f,c^2=f,d^2=ef,d^a=def,g^a=g^4,c^b=cf65 conjugacy classes — view the full character table.
| maximal subgroup of | #10599045, #10599046, #10599049, … |
| maximal normal subgroup of | #10599045, #10599046, #10599049, … |
| maximal quotient of | #10605371, #10605382, #10605391, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 1242);