Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| automorphism | Aut(G) = C2×#66297 (order 640) · Inn(G) = (C24⋊C5)⋊C2 · Out(G) = C22 |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Fitting | 2−1+4 ⋊ D5 |
| derived | 2−1+4⋊C5 ⋊ C2 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, hypercenter | C2 | (C24⋊C5)⋊C2 |
| commutator, cosocle, hypocenter | 2−1+4⋊C5 | C2 |
| Fitting | 2−1+4 | D5 |
| LCS | G ▷ 2−1+4⋊C5 | |
| UCS | C1 ◁ C2 | |
52 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,15,23,19)(2,5,27,24)(3,20,25,12)(4,6,28,32)(7,14,16,17)(8,13,26,31)(9,29,18,11)(10,21,30,22);(2,7,17,32,21)(3,12,26,22,9)(4,16,29,30,28)(5,10,13,8,23)(11,18,14,24,20)(15,19,31,25,27)Polycyclic presentation:
2,5,2^5|d^2=g,e^2=g,b^a=b^4,c^a=cd,d^a=dg,e^a=cdf,f^a=ce,c^b=cd,d^b=de,e^b=ef,f^b=cg,d^c=dg,e^c=eg,f^c=fg,e^d=eg14 conjugacy classes — view the full character table.
| maximal subgroup of | #10619373, C2×#66244, #10619380, … |
| maximal normal subgroup of | #10619373, C2×#66244, #10619380, … |
| maximal quotient of | #10617013, #10617014, #10617016, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 1582);