Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C5×#131 ⋊ C2 |
| Sylow | C5 ⋊ #578 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C22×D20 |
| commutator, cosocle | C4×C5 | C24 |
| Frattini | C4 | C23×D5 |
| Fitting | C5×#131 | C2 |
| socle | C2×C5 | C22×D4 |
| hypercenter | #131 | D5 |
| hypocenter | C5 | #578 |
| LCS | G ▷ C4×C5 ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C2 ◁ #41 ◁ #131 | |
248 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 4 generators on 37 points:
(1,2,4,5,3)(6,10,9,21,12,24,11,23)(7,15,25,35,16,30,14,22)(8,18,20,34,19,33,17,31)(13,27,26,36,29,37,28,32);(6,9,12,11)(7,14,16,25)(8,17,19,20)(10,21,24,23)(13,26,29,28)(15,22,30,35)(18,31,33,34)(27,36,37,32);(6,8,12,19)(7,13,16,29)(9,20,11,17)(10,18,24,33)(14,28,25,26)(15,27,30,37)(21,34,23,31)(22,32,35,36);(2,3)(4,5)(6,7)(8,13)(9,14)(10,22)(11,25)(12,16)(15,23)(17,26)(18,32)(19,29)(20,28)(21,30)(24,35)(27,31)(33,36)(34,37)Polycyclic presentation:
2^6,5|b^2=f,c^2=f,d^2=ef,e^2=f,d^a=de,e^a=ef,g^a=g^4,c^b=cf62 conjugacy classes — view the full character table.
| maximal subgroup of | #10599920, #10599922, #10601888, … |
| maximal normal subgroup of | #10599920, #10599922, #10601888, … |
| maximal quotient of | #10610277, #10610293, #10610299, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 1425);