Finite Group DatabasebetaFinite Group Databasebeta| order | 480 |
| isoclinism | isoclinic class of #91885 |
| automorphism | Aut(G) = C2×AGL(1,𝔽5)×S4 (order 960) · Inn(G) = A4⋊AGL(1,𝔽5) · Out(G) = C22 |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Fitting | C5×Q8 ⋊ Dic3 |
| derived | C5×SL(2,𝔽3) ⋊ C4 |
| Sylow | C5 ⋊ #859 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | A4⋊AGL(1,𝔽5) |
| commutator, hypocenter | C5×SL(2,𝔽3) | C4 |
| Fitting | C5×Q8 | Dic3 |
| socle | C2×C5 | SL(2,ℤ/4) |
| prime radical, cosocle | D5×SL(2,𝔽3) | C2 |
| LCS | G ▷ C5×SL(2,𝔽3) | |
| UCS | C1 ◁ C2 | |
56 conjugacy classes of subgroups.
| minimal degree | μ(G) = 13 |
| transitive degree | μt(G) = 40 |
A minimal faithful permutation representation — 2 generators on 13 points:
(1,2,8,3,4,7,6,5)(10,11,12,13);(2,5,8)(3,6,7)(9,10,13,11,12)Polycyclic presentation:
2^2,3,5,2^3|a^2=b,e^2=g,f^2=g,c^a=c^2,d^a=d^3,e^a=f,f^a=e,d^b=d^4,e^c=fg,f^c=ef,f^e=fg23 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,𝔽5)×#226, #11756704, #11756705, … |
| maximal normal subgroup of | AGL(1,𝔽5)×#226, #11756704, #11756705, … |
| maximal quotient of | SL(2,𝔽3)⋊#170, C2×#91886, #13085141, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(480, 962);