Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C5×#125 ⋊ C2 |
| Sylow | C5 ⋊ #541 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D5 |
| commutator, socle, cosocle | C22×C5 | C24 |
| Fitting | C5×#125 | C2 |
| prime radical | C22×D5 | C23 |
| hypercenter | #125 | D5 |
| hypocenter | C5 | #541 |
| LCS | G ▷ C22×C5 ▷ C5 | |
| UCS | C1 ◁ C22 ◁ #125 | |
192 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 4 generators on 29 points:
(1,2,4,5,3)(6,8,9,13)(7,11,10,12)(15,21)(17,26)(19,28)(22,25);(6,7,9,10)(8,12,13,11)(14,17)(15,20)(16,22)(18,26)(19,27)(21,24)(23,25)(28,29);(14,16,18,23)(15,19,21,28)(17,25,26,22)(20,29,24,27);(2,3)(4,5)(7,10)(11,12)(14,15)(16,19)(17,24)(18,21)(20,26)(22,29)(23,28)(25,27)Polycyclic presentation:
2^6,5|b^2=e,c^2=ef,d^2=ef,c^a=cf,d^a=de,g^a=g^4,c^b=ce,d^c=df50 conjugacy classes — view the full character table.
| maximal subgroup of | #10612876, #10612878, #10612914, … |
| maximal normal subgroup of | #10612876, #10612878, #10612914, … |
| maximal quotient of | #10605883, #10605904, #10605924, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 1369);