Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| isoclinism | isoclinic class of #64945 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C5 ⋊ #395 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C22×D5 |
| commutator, socle, prime radical, cosocle | C23×C5 | C23 |
| Fitting | C22×C5×Q8 | C2 |
| hypercenter | C22×Q8 | D5 |
| hypocenter | C5 | #395 |
| LCS | G ▷ C23×C5 ▷ C5 | |
| UCS | C1 ◁ C23 ◁ C22×Q8 | |
182 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 3 generators on 21 points:
(1,2,4,5,3)(7,9)(14,17,18,15)(16,19,20,21);(11,13)(14,16,18,20)(15,19,17,21);(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15,18,17)(16,19,20,21)Polycyclic presentation:
2^6,5|a^2=f,b^2=f,c^2=f,b^a=be,c^a=cd,g^a=g^4,c^b=cf62 conjugacy classes — view the full character table.
| maximal subgroup of | #10598830, #10601767, #10601768, … |
| maximal normal subgroup of | #10598830, #10601767, #10601768, … |
| maximal quotient of | #10602359, #10602361, #10602532, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 856);