Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C4×C5×Q8 ⋊ C2 |
| Sylow | C5 ⋊ #520 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D5 |
| commutator | C2×C5 | C23×C4 |
| Fitting | C4×C5×Q8 | C2 |
| socle, cosocle | C22×C5 | C24 |
| prime radical | C2×#713 | C2 |
| hypercenter | C4×Q8 | D5 |
| hypocenter | C5 | #520 |
| LCS | G ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C22 ◁ C4×Q8 | |
294 conjugacy classes of subgroups.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 4 generators on 17 points:
(1,2,4,5,3)(6,8,9,7)(10,12,13,16)(11,15,14,17);(6,7,9,8)(11,14)(15,17);(10,11,13,14)(12,15,16,17);(2,3)(4,5)(12,16)(15,17)Polycyclic presentation:
2^6,5|b^2=e,c^2=ef,d^2=f,d^a=de,g^a=g^4,c^b=ce74 conjugacy classes — view the full character table.
| maximal subgroup of | #10611378, #10611381, #10611428, … |
| maximal normal subgroup of | #10611378, #10611381, #10611428, … |
| maximal quotient of | #10605357, #10605360, #10605364, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 1246);