Finite Group DatabasebetaFinite Group Databasebeta| order | 360 |
| isoclinism | isoclinic class of #67195 |
| automorphism | Aut(G) = AGL(1,𝔽5)×AGL(1,𝔽9) (order 1440) · Inn(G) = G · Out(G) = C4 |
| homology | Abelianization (H1) = C8 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Fitting, socle, derived, Hall | C32×C5 ⋊ C8 |
| Sylow | C32 ⋊ #170 |
| Sylow | C5 ⋊ AGL(1,𝔽9) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C32×C5 | C8 |
| prime radical | C5×C3⋊D3 | C4 |
| cosocle | #4379 | C2 |
| LCS | G ▷ C32×C5 | |
| UCS | C1 | |
20 conjugacy classes of subgroups.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 45 |
A minimal faithful permutation representation — 2 generators on 14 points:
(2,4,5,7,6,3,8,9)(11,12,13,14);(1,7,9)(2,8,4)(3,5,6)(10,11,12,14,13)Polycyclic presentation:
2^3,3^2,5|a^2=b,b^2=c,d^a=e^2,e^a=d^2e,f^a=f^2,d^b=de^2,e^b=d^2e^2,f^b=f^4,d^c=d^2,e^c=e^215 conjugacy classes — view the full character table.
| maximal subgroup of | D5⋊AGL(1,𝔽9), C2×#67195, C3×#67195, … |
| maximal normal subgroup of | D5⋊AGL(1,𝔽9), C2×#67195, C3×#67195, … |
| maximal quotient of | #10623955, C2×#67195, #11761921, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(360, 125);