Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C5 ⋊ #383 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C22×D5 |
| commutator | C22×C5 | C22×C4 |
| Fitting | C2×C42×C5 | C2 |
| socle, prime radical, cosocle | C23×C5 | C23 |
| hypercenter | C2×C42 | D5 |
| hypocenter | C5 | #383 |
| LCS | G ▷ C22×C5 ▷ C5 | |
| UCS | C1 ◁ C23 ◁ C2×C42 | |
138 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 3 generators on 25 points:
(1,2,4,5,3)(11,15)(14,17)(18,20,21,25)(19,23,22,24);(10,12,13,16)(11,14,15,17)(19,22)(23,24);(2,3)(4,5)(6,7,8,9)(10,11)(12,14)(13,15)(16,17)(18,19)(20,24)(21,22)(23,25)Polycyclic presentation:
2^6,5|a^2=d,b^2=e,c^2=f,b^a=bf,c^a=cef,g^a=g^492 conjugacy classes — view the full character table.
| maximal subgroup of | #10598157, #10598243, #10601584, … |
| maximal normal subgroup of | #10598157, #10598243, #10601584, … |
| maximal quotient of | #10600186, #10600209, #10600231, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 564);