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 | C5×#95 | C2 |
| socle, prime radical, cosocle | C23×C5 | C23 |
| hypercenter | #95 | D5 |
| hypocenter | C5 | #383 |
| LCS | G ▷ C22×C5 ▷ C5 | |
| UCS | C1 ◁ C23 ◁ #95 | |
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)(6,7,8,9)(11,15)(14,17)(20,24)(22,25);(10,12,13,16)(11,14,15,17)(18,20,21,24)(19,22,23,25);(2,3)(4,5)(10,11)(12,14)(13,15)(16,17)(18,19,21,23)(20,22,24,25)Polycyclic presentation:
2^6,5|a^2=f,b^2=e,c^2=d,c^a=cef,g^a=g^4,c^b=cf68 conjugacy classes — view the full character table.
| maximal subgroup of | #10598505, #10599074, #10599076, … |
| maximal normal subgroup of | #10598505, #10599074, #10599076, … |
| maximal quotient of | #10599400, #10599401, #10599489, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 278);